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<p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/almost+symplectic+structure">almost symplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metaplectic+structure">metaplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metalinear+structure">metalinear structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>, <a class="existingWikiWord" href="/nlab/show/n-plectic+form">n-plectic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> <p><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/symplectomorphism+group">symplectomorphism group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+vector+field">symplectic vector field</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+form">Hamiltonian form</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+gradient">symplectic gradient</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a>, <a class="existingWikiWord" href="/nlab/show/moment+map">moment map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+formalism">BRST-BV formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>, <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> </li> </ul> <h2 id="classical_mechanics_and_quantization">Classical mechanics and quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>,</p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong>, <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+form">contact form</a>, <a class="existingWikiWord" href="/nlab/show/Reeb+vector+field">Reeb vector field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+n-algebra">Heisenberg Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a></p> </li> </ul> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/symplectic+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#various'>Various</a></li> <li><a href='#PresymplecticManifolds'>Pre-symplectic manifolds and infinitesimal quantomorphisms</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#deformation_quantization'>Deformation quantization</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Poisson manifolds are a mathematical setup for <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a> with finitely many degrees of freedom.</p> <h2 id="definition">Definition</h2> <p>A <strong><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a></strong> is a commutative unital <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, in this case over the field of <a class="existingWikiWord" href="/nlab/show/real+number">real</a> or <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, equipped with a <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>,</mo><mo stretchy="false">}</mo><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\{,\} \colon A\otimes A\to A</annotation></semantics></math> such that, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f\in A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\{ f,-\} \colon A\to A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as an associative algebra.</p> <p>A <strong>Poisson manifold</strong> is a real <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> equipped with a <strong>Poisson structure</strong>. A Poisson structure is a <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>,</mo><mo stretchy="false">}</mo><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\{,\} \colon C^\infty(M)\times C^\infty(M)\to C^\infty(M)</annotation></semantics></math> on the vector space of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> which together with the pointwise multiplication of functions makes it a Poisson algebra. As derivations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(M)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">correspond</a> to smooth <a class="existingWikiWord" href="/nlab/show/tangent+vector+fields">tangent vector fields</a>, for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f\in C^\infty(M)</annotation></semantics></math> there is a vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">X_f</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X_f(g)=\{f,g\}</annotation></semantics></math> and called the <strong><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></strong> corresponding to the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, which is viewed as a classical hamiltonian function.</p> <p>Alternatively a Poisson structure on a manifold is given by a choice of smooth antisymmetric <a class="existingWikiWord" href="/nlab/show/bivector">bivector</a> called a <strong>Poisson bivector</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∈</mo><msup><mi>Λ</mi> <mn>2</mn></msup><mi>T</mi><mi>M</mi></mrow><annotation encoding="application/x-tex">P\in\Lambda^2 T M</annotation></semantics></math>; then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">}</mo><mo>:</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>d</mi><mi>f</mi><mo>⊗</mo><mi>d</mi><mi>g</mi><mo>,</mo><mi>P</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\{f,g\}:=\langle d f\otimes d g, P\rangle</annotation></semantics></math>.</p> <p>This induces and is equivalently encoded by the structure of a <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>M</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">h \colon M\to N</annotation></semantics></math> of Poisson manifolds is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> between the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> which preserves the Poisson brackets, in that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f, g \in C^\infty(N)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>f</mi><mo>∘</mo><mi>h</mi><mo>,</mo><mi>g</mi><mo>∘</mo><mi>h</mi><msub><mo stretchy="false">}</mo> <mi>M</mi></msub><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><msub><mo stretchy="false">}</mo> <mi>N</mi></msub></mrow><annotation encoding="application/x-tex">\{f\circ h, g\circ h\}_M = \{f,g\}_N</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <h3 id="various">Various</h3> <div class="num_example"> <h6 id="example">Example</h6> <p>Every manifold admits the <em>trivial Poisson structure</em> for which the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> simply vanishes on all elements.</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> carries a natural Poisson structure. Given a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f: X \to \mathbb{R}</annotation></semantics></math>, the symplectic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> induces <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>f</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">H_f: X \to T X</annotation></semantics></math>. We can then define the Poisson bracket as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">}</mo><mo>=</mo><msub><mi>H</mi> <mi>f</mi></msub><mi>g</mi><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>f</mi></msub><mo>,</mo><msub><mi>H</mi> <mi>g</mi></msub><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{f, g\} = H_f g = \omega(H_f, H_g). </annotation></semantics></math></div> <p>See <a href="PresymplecticManifolds">below</a> for more. However, such Poisson manifolds are very special. It is a basic theorem that Poisson structures on a manifold are equivalent to the smooth <a class="existingWikiWord" href="/nlab/show/foliations">foliations</a> of the underlying manifold such that each leaf is a symplectic manifold.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>The dual to a finite-dimensional <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> has a natural structure of a Poisson manifold, the <em><a class="existingWikiWord" href="/nlab/show/Lie-Poisson+structure">Lie-Poisson structure</a></em>. Its leaves are called <a class="existingWikiWord" href="/nlab/show/coadjoint+orbits">coadjoint orbits</a>.</p> </div> <div class="num_example"> <h6 id="example_4">Example</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> and given a <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">H \colon X \longrightarrow \mathbb{R}</annotation></semantics></math>, there is a Poisson bracket on the functions on the <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth</a> <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[I,X]</annotation></semantics></math> (the “space of histories” or “space of <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a>”), for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I = [0,1]</annotation></semantics></math> the closed <a class="existingWikiWord" href="/nlab/show/interval">interval</a>, which is such that its <a class="existingWikiWord" href="/nlab/show/symplectic+leaves">symplectic leaves</a> are each a copy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, but regarded as the space of initial conditions for evolution with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/source">source</a> term added. For more on this see at <em><a class="existingWikiWord" href="/nlab/show/off-shell+Poisson+bracket">off-shell Poisson bracket</a></em>.</p> </div> <div class="num_example"> <h6 id="example_5">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/local+action+functional">local action functional</a> which admits a <a class="existingWikiWord" href="/nlab/show/Green%27s+function">Green's function</a> for its <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> defines the <a class="existingWikiWord" href="/nlab/show/Peierls+bracket">Peierls bracket</a> on <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> (where in fact it is symplectic) and also “<a class="existingWikiWord" href="/nlab/show/off-shell+Poisson+bracket">off-shell</a>” on all of configuration space, where it is a genuine Poisson bracket, the canonocal Poisson bracket of the corresponding <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a>.</p> </div> <h3 id="PresymplecticManifolds">Pre-symplectic manifolds and infinitesimal quantomorphisms</h3> <p>We discuss the traditional definition of the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> of a (<a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">pre-</a>)<a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>, and then show how it may equivalently be understood as the algebra of infinitesimal symmetries of any of the <a class="existingWikiWord" href="/nlab/show/prequantizations">prequantizations</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>. For more on this see at <em><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+prequantum+geometry">geometry of physics – prequantum geometry</a></em>.</p> <div class="num_defn" id="PresymplecticManifold"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>. A closed <a class="existingWikiWord" href="/nlab/show/differential+2-form">differential 2-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega_{cl}^2(X)</annotation></semantics></math> is a <em><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a></em> if it is non-degenerate in that the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of the operation of contracting with <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mi>ω</mi><mo lspace="verythinmathspace">:</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \iota_{(-)}\omega \colon Vect(X) \longrightarrow \Omega^1(X) </annotation></semantics></math></div> <p>is trivial: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>ι</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">ker(\iota_{(-)}\omega) = 0</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is just closed with possibly non-trivial kernel, we call it a <a class="existingWikiWord" href="/nlab/show/presymplectic+form">presymplectic form</a>. (We do not require here the dimension of the kernel restricted to each tangent space to be constant.)</p> </div> <div class="num_defn" id="HamiltonianVectorField"> <h6 id="definition_3">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">presymplectic manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \omega)</annotation></semantics></math>, then a <em><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a></em> for a <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in Vect(X)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \in C^\infty(X)</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>v</mi></msub><mi>ω</mi><mo>+</mo><mi>d</mi><mi>H</mi><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \iota_{v} \omega + d H = 0 \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in Vect(X)</annotation></semantics></math> is such that there <em>exists</em> at least one Hamiltonian for it then it is called a <em><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></em>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>HamVect</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> HamVect(X,\omega) \hookrightarrow Vect(X) </annotation></semantics></math></div> <p>for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+subspace">linear subspace</a> of Hamiltonian vector fields among all <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a></p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is symplectic then, evidently, there is a unique Hamiltonian vector field, def. <a class="maruku-ref" href="#HamiltonianVectorField"></a>, associated with every Hamiltonian, i.e. every smooth function is then the Hamiltonian of precisely one Hamiltonian vector field (but two different Hamiltonians may still have the same Hamiltonian vector field uniquely associated with them). As far as <a class="existingWikiWord" href="/nlab/show/prequantum+geometry">prequantum geometry</a> is concerned, this is all that the non-degeneracy condition that makes a closed 2-form be symplectic is for. But we will see that the definitions of <a class="existingWikiWord" href="/nlab/show/Poisson+brackets">Poisson brackets</a> and of <a class="existingWikiWord" href="/nlab/show/quantomorphism+groups">quantomorphism groups</a> directly generalize also to the presymplectic situation, simply by considering not just Hamiltonian functions but pairs of a Hamiltonian vector field and a compatible Hamiltonian.</p> </div> <div class="num_defn" id="PoissonBracket"> <h6 id="definition_4">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">presymplectic manifold</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ham</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>HamVect</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ham(X,\omega) \hookrightarrow HamVect(X,\omega) \oplus C^\infty(X) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/linear+subspace">linear subspace</a> of the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a>, def. <a class="maruku-ref" href="#HamiltonianVectorField"></a>, and <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on those pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v,H)</annotation></semantics></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ham</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mi>ι</mi> <mi>v</mi></msub><mi>ω</mi><mo>+</mo><mi>d</mi><mi>H</mi><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ham(X,\omega) \coloneqq \left\{ (v,H) | \iota_v \omega + d H = 0 \right\} \,. </annotation></semantics></math></div> <p>Define a <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ham</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>Ham</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Ham</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [-,-] \;\colon\; Ham(X,\omega) \otimes Ham(X,\omega) \longrightarrow Ham(X,\omega) </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≔</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mi>ω</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [(v_1,H_1), (v_2,H_2)] \coloneqq ([v_1,v_2], \iota_{v_2}\iota_{v_1} \omega) \,, </annotation></semantics></math></div> <p>called the <em><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a></em>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[v_1,v_2]</annotation></semantics></math> is the standard Lie bracket on <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔓𝔬𝔦𝔰</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>Ham</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{Pois}(X,\omega) \coloneqq (Ham(X,\omega),[-,-]) </annotation></semantics></math></div> <p>for the resulting <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>. In the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is symplectic, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ham</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ham(X,\omega) \simeq C^\infty(X)</annotation></semantics></math> and hence in this case</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔓𝔬𝔦𝔰</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{Pois}(X,\omega) \simeq (C^\infty(X),[-,-]) \,. </annotation></semantics></math></div></div> <div class="num_example" id="HeisenbergAlgebraOfSymplecticVectorSpace"> <h6 id="example_6">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">X = \mathbb{R}^{2n}</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><mi>d</mi><msub><mi>p</mi> <mi>i</mi></msub><mo>∧</mo><mi>d</mi><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\omega = \sum_{i = 1}^n d p_i \wedge d q^i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>q</mi> <mi>i</mi></msup><msubsup><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\{q^i\}_{i = 1}^n</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/canonical+coordinates">canonical coordinates</a> on one copy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>p</mi> <mi>i</mi></msub><msubsup><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\{p_i\}_{i = 1}^n</annotation></semantics></math> that on the other (“<a class="existingWikiWord" href="/nlab/show/canonical+momenta">canonical momenta</a>”). Hence let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2n</annotation></semantics></math>, regarded as a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>.</p> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Vect(X)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/linear+span">spanned</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> by the canonical bases vector fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mo>∂</mo> <mrow><msup><mi>q</mi> <mi>i</mi></msup></mrow></msub><mo>,</mo><msub><mo>∂</mo> <mrow><msup><mi>p</mi> <mi>i</mi></msup></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\partial_{q^i}, \partial_{p^i}\}</annotation></semantics></math>. These basis vector fields are manifestly <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a> via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mrow><msup><mi>q</mi> <mi>i</mi></msup></mrow></msub></mrow></msub><mi>ω</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>d</mi><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> \iota_{\partial_{q^i}} \omega = - d p_i </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow></msub></mrow></msub><mi>ω</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>d</mi><msup><mi>q</mi> <mi>i</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \iota_{\partial_{p_i}} \omega = + d q^i \,. </annotation></semantics></math></div> <p>Moreover, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a>, these Hamiltonians are unique up to a choice of constant function. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>i</mi></mstyle><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{i} \in C^\infty(X)</annotation></semantics></math> for the unit constant function, then the nontrivial Poisson brackets between the basis vector fields are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>q</mi> <mi>i</mi></msup><mo>,</mo><msub><mi>p</mi> <mi>j</mi></msub><mo stretchy="false">]</mo><mo>≔</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo>∂</mo> <mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow></msub><mo>,</mo><msup><mi>q</mi> <mi>i</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mo>∂</mo> <mrow><msup><mi>q</mi> <mi>j</mi></msup></mrow></msub><mo>,</mo><msub><mi>p</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mi>δ</mi> <mi>j</mi> <mi>i</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mstyle mathvariant="bold"><mi>i</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mi>δ</mi> <mi>j</mi> <mi>i</mi></msubsup><mstyle mathvariant="bold"><mi>i</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [q^i, p_j] \coloneqq [(-\partial_{p_i}, q^i), (\partial_{q^j}, p_j)] = - \delta_j^i (0, \mathbf{i}) = - \delta_j^i \mathbf{i} \,. </annotation></semantics></math></div> <p>This is called the <a class="existingWikiWord" href="/nlab/show/Heisenberg+algebra">Heisenberg algebra</a>.</p> <p>More generally, the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a> corresponding to <a class="existingWikiWord" href="/nlab/show/quadratic+Hamiltonians">quadratic Hamiltonians</a>, i.e. degree-2 <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>q</mi> <mi>i</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{q^i\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{p_i\}</annotation></semantics></math>, generate the <a class="existingWikiWord" href="/nlab/show/affine+symplectic+group">affine symplectic group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>. The freedom to add constant terms to Hamiltonians gives the <a class="existingWikiWord" href="/nlab/show/extended+affine+symplectic+group">extended affine symplectic group</a>.</p> </div> <p>Example <a class="maruku-ref" href="#HeisenbergAlgebraOfSymplecticVectorSpace"></a> serves to motivate a more conceptual origin of the definition of the Poisson bracket in def. <a class="maruku-ref" href="#PoissonBracket"></a>.</p> <div class="num_example"> <h6 id="example_7">Example</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>≔</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><msub><mi>p</mi> <mi>i</mi></msub><mi>d</mi><msub><mi>q</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \theta \coloneqq \sum_{i = 1}^n p_i d q_i \in \Omega^1(\mathbb{R}^{2n}) </annotation></semantics></math></div> <p>for the canonical choice of <a class="existingWikiWord" href="/nlab/show/differential+1-form">differential 1-form</a> satisfying</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>θ</mi><mo>=</mo><mi>ω</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d \theta = \omega \,. </annotation></semantics></math></div> <p>If we regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>≃</mo><msup><mi>T</mi> <mo>*</mo></msup><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2n} \simeq T^\ast \mathbb{R}^n</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, then this is what is known as the <a class="existingWikiWord" href="/nlab/show/Liouville-Poincar%C3%A9+1-form">Liouville-Poincaré 1-form</a>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2n}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/contractible+topological+space">contractible</a> as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, every <a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a> over it is necessarily trivial, and hence any choice of 1-form such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> may canonically be thought of as being a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> on the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>. As such this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/prequantization">prequantization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>,</mo><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><mi>d</mi><msub><mi>p</mi> <mi>i</mi></msub><mo>∧</mo><mi>d</mi><msup><mi>q</mi> <mi>i</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{R}^{2n}, \sum_{i=1}^n d p_i \wedge d q^i)</annotation></semantics></math>.</p> <p>Being thus a <a class="existingWikiWord" href="/nlab/show/circle+bundle+with+connection">circle bundle with connection</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> has more symmetry than its <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> has: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \in C^\infty(\mathbb{R}^{2n}, U(1))</annotation></semantics></math> any smooth function, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>↦</mo><mi>θ</mi><mo>+</mo><mi>d</mi><mi>α</mi></mrow><annotation encoding="application/x-tex"> \theta \mapsto \theta + d\alpha </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math>, leading to a different but equivalent <a class="existingWikiWord" href="/nlab/show/prequantization">prequantization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>.</p> <p>Hence while a <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> is said to preserve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> (is a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+field">symplectic vector field</a>) if the <a class="existingWikiWord" href="/nlab/show/Lie+derivative">Lie derivative</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> vanishes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>v</mi></msub><mi>ω</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \mathcal{L}_v \omega = 0 </annotation></semantics></math></div> <p>in the presence of a choice for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> the right condition to ask for is that there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>v</mi></msub><mi>θ</mi><mo>=</mo><mi>d</mi><mi>α</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{L}_v \theta = d \alpha \,. </annotation></semantics></math></div></div> <p>For more on this see also at <em><a class="existingWikiWord" href="/nlab/show/prequantized+Lagrangian+correspondence">prequantized Lagrangian correspondence</a></em>.</p> <p>Notice then the following basic but important fact.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">presymplectic manifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta \in \Omega^1(X)</annotation></semantics></math> a 1-form such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>θ</mi><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">d \theta = \omega</annotation></semantics></math> then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v,\alpha) \in Vect(X)\oplus C^\infty(X)</annotation></semantics></math> the condition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>v</mi></msub><mi>θ</mi><mo>=</mo><mi>d</mi><mi>α</mi></mrow><annotation encoding="application/x-tex"> \mathcal{L}_v \theta = d \alpha </annotation></semantics></math></div> <p>is equivalent to the condition that makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>≔</mo><msub><mi>ι</mi> <mi>v</mi></msub><mi>θ</mi><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">H \coloneqq \iota_v \theta - \alpha </annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> according to def. <a class="maruku-ref" href="#HamiltonianVectorField"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>v</mi></msub><mi>ω</mi><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>v</mi></msub><mi>θ</mi><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \iota_v \omega + d (\iota_v \theta - \alpha ) = 0 \,. </annotation></semantics></math></div> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>, def. <a class="maruku-ref" href="#PoissonBracket"></a>, between two such Hamiltonian pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>α</mi> <mi>i</mi></msub><mo>−</mo><msub><mi>ι</mi> <mi>v</mi></msub><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_i, \alpha_i -\iota_v \theta)</annotation></semantics></math> is equivalently given by the skew-symmetric <a class="existingWikiWord" href="/nlab/show/Lie+derivative">Lie derivative</a> of the corresponding vector fields on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_i</annotation></semantics></math>:</p> <div class="maruku-equation" id="eq:EquationForLieHomomorphism"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow></msub><mi>θ</mi><mo>−</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mi>ω</mi><mo>=</mo><msub><mi>ℒ</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><msub><mi>α</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>ℒ</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><msub><mi>α</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> \iota_{[v_1,v_2]} \theta - \iota_{v_2}\iota_{v_1}\omega = \mathcal{L}_{v_1} \alpha_2 - \mathcal{L}_{v_2} \alpha_1 </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Using <a class="existingWikiWord" href="/nlab/show/Cartan%27s+magic+formula">Cartan's magic formula</a> and by the prequantization condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>θ</mi><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">d \theta = \omega</annotation></semantics></math> the we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>ℒ</mi> <mi>v</mi></msub><mi>θ</mi></mtd> <mtd><mo>=</mo><msub><mi>ι</mi> <mi>v</mi></msub><mi>d</mi><mi>θ</mi><mo>+</mo><mi>d</mi><msub><mi>ι</mi> <mi>v</mi></msub><mi>θ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>ι</mi> <mi>v</mi></msub><mi>ω</mi><mo>+</mo><mi>d</mi><msub><mi>ι</mi> <mi>v</mi></msub><mi>θ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathcal{L}_v \theta &= \iota_v d\theta + d \iota_v \theta \\ & = \iota_v\omega + d \iota_v \theta \end{aligned} \,. </annotation></semantics></math></div> <p>This gives the first statement. For the second we first use the formula for the <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> and then again the definition of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_i</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mi>ω</mi></mtd> <mtd><mo>=</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mi>d</mi><mi>θ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mi>d</mi><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><mi>θ</mi><mo>−</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><mi>d</mi><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mi>θ</mi><mo>−</mo><msub><mi>ι</mi> <mrow><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow></msub><mi>θ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mi>d</mi><msub><mi>α</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><mi>ω</mi><mo>−</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><mi>d</mi><msub><mi>α</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mi>ω</mi><mo>−</mo><msub><mi>ι</mi> <mrow><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow></msub><mi>θ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>2</mn><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mi>ω</mi><mo>+</mo><msub><mi>ℒ</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><msub><mi>α</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>ℒ</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><msub><mi>α</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>ι</mi> <mrow><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow></msub><mi>θ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \iota_{v_2}\iota_{v_1} \omega & = \iota_{v_2}\iota_{v_1} d\theta \\ & = \iota_{v_1} d \iota_{v_2} \theta - \iota_{v_2} d \iota_{v_1} \theta - \iota_{[v_1,v_2]} \theta \\ & = \iota_{v_1} d \alpha_2 - \iota_{v_1} \iota_{v_2}\omega - \iota_{v_2} d \alpha_1 + \iota_{v_2} \iota_{v_1}\omega - \iota_{[v_1,v_2]} \theta \\ & = 2 \iota_{v_2} \iota_{v_1}\omega + \mathcal{L}_{v_1} \alpha_2 -\mathcal{L}_{v_2} \alpha_1 - \iota_{[v_1,v_2]} \theta \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_cor" id="EquivalenceBetweenPoissonBracketAndInfinQuantomorphism"> <h6 id="corollary">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">presymplectic manifold</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta \in \Omega^1(X)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>θ</mi><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">d \theta = \omega</annotation></semantics></math>, consider the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔔𝔲𝔞𝔫𝔱𝔐𝔬𝔯𝔭𝔥</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mi>ℒ</mi> <mi>v</mi></msub><mi>θ</mi><mo>=</mo><mi>d</mi><mi>α</mi><mo>}</mo></mrow><mo>⊂</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{QuantMorph}(X,\theta) = \left\{ (v,\alpha) | \mathcal{L}_v \theta = d \alpha \right\} \subset Vect(X) \oplus C^\infty(X) </annotation></semantics></math></div> <p>with Lie bracket</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>α</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>α</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>ℒ</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><msub><mi>α</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>ℒ</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><msub><mi>α</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [(v_1,\alpha_1), (v_2,\alpha_2)] = ([v_1,v_2], \mathcal{L}_{v_1}\alpha_2 - \mathcal{L}_{v_2}\alpha_1) \,. </annotation></semantics></math></div> <p>Then by <a class="maruku-eqref" href="#eq:EquationForLieHomomorphism">(1)</a> the linear map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><msub><mi>ι</mi> <mi>v</mi></msub><mi>θ</mi><mo>−</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (v,H) \mapsto (v, \iota_v \theta - H) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔓𝔬𝔦𝔰</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>𝔔𝔲𝔞𝔫𝔱𝔐𝔬𝔯𝔭𝔥</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{Pois}(X,\omega) \stackrel{\simeq}{\longrightarrow} \mathfrak{QuantMorph}(X,\theta) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+algebra">Poisson bracket Lie algebra</a>, def. <a class="maruku-ref" href="#PoissonBracket"></a>.</p> </div> <p>This shows that for exact pre-symplectic forms the Poisson bracket Lie algebra is secretly the Lie algebra of infinitesimal symmetries of any of its <a class="existingWikiWord" href="/nlab/show/prequantizations">prequantizations</a>. In fact this holds true also when the pre-symplectic form is not exact:</p> <div class="num_defn" id="CechDelignePrequantizationAnditsInfinitesimalAutomorpisms"> <h6 id="definition_5">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">presymplectic manifold</a>, a <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a>-<a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne</a> <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>θ</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\{U_i\},\{g_{i j}, \theta_i\})</annotation></semantics></math> for a <em><a class="existingWikiWord" href="/nlab/show/prequantization">prequantization</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\{U_i \to X\}_i</annotation></semantics></math>;</p> </li> <li> <p>1-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>θ</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\theta_i \in \Omega^1(U_i)\}</annotation></semantics></math>;</p> </li> <li> <p>smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{g_{i j} \in C^\infty(U_{i j}, U(1))\}</annotation></semantics></math></p> </li> </ol> <p>such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>θ</mi> <mi>i</mi></msub><mo>=</mo><mi>ω</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">d \theta_i = \omega|_{U_i}</annotation></semantics></math> on all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>θ</mi> <mi>i</mi></msub><mo>+</mo><mi>d</mi><mi>log</mi><msub><mi>g</mi> <mi>ij</mi></msub></mrow><annotation encoding="application/x-tex">\theta_j = \theta_i + d log g_{ij}</annotation></semantics></math> on all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">U_{i j}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>g</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">g_{i j} g_{j k} = g_{i k}</annotation></semantics></math> on all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">U_{i j k}</annotation></semantics></math>.</p> </li> </ol> <p>The <em>quantomorphism Lie algebra</em> of this is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔔𝔲𝔞𝔫𝔱𝔐𝔬𝔯𝔭𝔥</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>θ</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>α</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mi>ℒ</mi> <mi>v</mi></msub><mi>log</mi><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>α</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>α</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><msub><mi>ℒ</mi> <mi>v</mi></msub><msub><mi>θ</mi> <mi>i</mi></msub><mo>=</mo><mi>d</mi><msub><mi>α</mi> <mi>i</mi></msub><mo>}</mo></mrow><mo>⊂</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo><mi>i</mi></munder><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathfrak{QuantMorph}(X,\{U_i\},\{g_{i j}, \theta_i\}) = \left\{ (v, \{\alpha_i\}) | \mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i \,, \mathcal{L}_v \theta_i = d \alpha_i \right\} \subset Vect(X) \oplus \left(\underset{i}{\bigoplus} C^\infty(U_i)\right) </annotation></semantics></math></div> <p>with bracket</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≔</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>ℒ</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>−</mo><msub><mi>ℒ</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [(v_1, \{(\alpha_1)_i\}), (v_2, \{(\alpha_2)_i\})] \coloneqq ([v_1,v_2], \{\mathcal{L}_{v_1}(\alpha_2)_i - \mathcal{L}_{v_2} (\alpha_1)_i\}) \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">presymplectic manifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>θ</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\{U_i\},\{g_{i j}, \theta_i\})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/prequantization">prequantization</a>, def. <a class="maruku-ref" href="#CechDelignePrequantizationAnditsInfinitesimalAutomorpisms"></a>, the linear map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>ι</mi> <mi>v</mi></msub><msub><mi>θ</mi> <mi>i</mi></msub><mo>−</mo><mi>H</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (v,H) \mapsto (v, \{\iota_v \theta_i - H|_{U_i}\}) </annotation></semantics></math></div> <p>constitutes an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔓𝔬𝔦𝔰</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>𝔔𝔲𝔞𝔫𝔱𝔐𝔬𝔯𝔭𝔥</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>θ</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{Pois}(X,\omega) \stackrel{\simeq}{\longrightarrow} \mathfrak{QuantMorph}(X,\{U_i\},\{g_{i j}, \theta_i\}) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℒ</mi> <mi>v</mi></msub><mi>log</mi><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>α</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>α</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i</annotation></semantics></math> on the infinitesimal quantomorphisms, togther with the Cech-Deligne cocycle condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>log</mi><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>θ</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>θ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d log g_{i j} = \theta_j - \theta_i</annotation></semantics></math> says that on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">U_{i j}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>v</mi></msub><msub><mi>θ</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>α</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>ι</mi> <mi>v</mi></msub><msub><mi>θ</mi> <mi>i</mi></msub><mo>−</mo><msub><mi>α</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> \iota_v \theta_j - \alpha_j = \iota_v \theta_i - \alpha_i </annotation></semantics></math></div> <p>and hence that there is a globally defined function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \in C^\infty(X)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>v</mi></msub><msub><mi>θ</mi> <mi>i</mi></msub><mo>−</mo><msub><mi>α</mi> <mi>i</mi></msub><mo>=</mo><mi>H</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\iota_v \theta_i - \alpha_i = H|_{U_i}</annotation></semantics></math>. This shows that the map is an isomrophism of vector spaces.</p> <p>Now over each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> the the situation for the brackets is just that of corollary <a class="maruku-ref" href="#EquivalenceBetweenPoissonBracketAndInfinQuantomorphism"></a> implied by <a class="maruku-eqref" href="#eq:EquationForLieHomomorphism">(1)</a>, hence the morphism is a Lie homomorphism.</p> </div> <h2 id="properties">Properties</h2> <h3 id="deformation_quantization">Deformation quantization</h3> <p><a class="existingWikiWord" href="/nlab/show/Kontsevich+formality">Kontsevich formality</a> implies that every Poisson manifold has a family of <a class="existingWikiWord" href="/nlab/show/deformation+quantizations">deformation quantizations</a>, parameterized by the <a class="existingWikiWord" href="/nlab/show/Grothendieck-Teichm%C3%BCller+group">Grothendieck-Teichmüller group</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+tensor">Poisson tensor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+geometry">Poisson geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+cohomology">Poisson cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+reduction">Poisson reduction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, <a class="existingWikiWord" href="/nlab/show/coisotropic+submanifold">coisotropic submanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+leaf">symplectic leaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+realization">symplectic realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+dual+pair">symplectic dual pair</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moyal+quantization">Moyal quantization</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Poisson+structure">higher Poisson structure</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Nambu+bracket">Nambu bracket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a></p> </li> </ul> </li> </ul> <p><br /></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/duality+between+algebra+and+geometry">duality between</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></strong></p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/category">category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dual+category">dual category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a></mtext></mover><msubsup><mi>Alg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a></mtext></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup><mo>,</mo><mi>comm</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">comm. C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncomm. topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCTopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">NCTopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>general <a class="existingWikiWord" href="/nlab/show/C-star-algebra">C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>Schemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}Schemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op} </annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A} \phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncomm. algebraic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCSchemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">NCSchemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mrow><mi>fin</mi><mo>,</mo><mi>red</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/finitely+generated+algebra">fin. gen.</a> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothManifolds</mi></mrow><annotation encoding="application/x-tex">SmoothManifolds</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mi>comm</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>SuperSpaces</mi> <mi>Cart</mi></msub></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mphantom><mtext>Pursell's theorem</mtext></mphantom></mover></mtd> <mtd><msubsup><mi>Alg</mi> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mphantom><mi>AAAA</mi></mphantom></mrow> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>q</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \overset{\phantom{\text{Pursell's theorem}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/formal+moduli+problem">formal</a> <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>(<a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super Lie theory</a>)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mi>Super</mi><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>fin</mi></msub></mtd></mtr> <mtr><mtd><mi>𝔤</mi></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><mtext><a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a></mtext><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><msup><mi>sdgcAlg</mi> <mi>op</mi></msup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">FDAs</a>”)</td></tr> </tbody></table> <p><strong>in <a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>:</p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/space+of+states">space of states</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/n-plectic+manifold">n-plectic manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/En-algebras">En-algebras</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">BD</a>-<a class="existingWikiWord" href="/nlab/show/BV+quantization">BV quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+algebra+of+observables">factorization algebra of observables</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+homology">factorization homology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism representation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div> <p><br /></p> <div> <p><strong>Examples of sequences of local structures</strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th>point</th><th>first order <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/formal+geometry">formal</a> = arbitrary order infinitesimal</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>local = <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>finite</th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <strong><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/integration">integration</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derivative">derivative</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Taylor+series">Taylor series</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a> (path)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tangent+vector">tangent vector</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/jet">jet</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a> of <a class="existingWikiWord" href="/nlab/show/curve">curve</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ+of+a+space">germ of a space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/square-0+ring+extension">square-0 ring extension</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nilpotent+ring+extension">nilpotent ring extension</a>/<a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring+extension">ring extension</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/finite+field">finite field</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{(p)}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">localization at (p)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/integers">integers</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+Lie+group">local Lie group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a></td><td style="text-align: left;"></td><td style="text-align: left;">local strict deformation quantization</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantization</a></td></tr> </tbody></table> </div> <p><br /></p> <h2 id="references">References</h2> <p>The notion is due to:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Lichnerowicz">André Lichnerowicz</a>, <em>Les variétés de Poisson et leurs algèbres de Lie associées</em>, Journal of Differential Geometry <strong>12</strong> 2 (1977) 253–300 [<a href="http://dx.doi.org/10.4310/jdg/1214433987">doi:10.4310/jdg/1214433987</a>]</li> </ul> <p>Further early discussion:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, <em>The local structure of Poisson manifolds</em>, J. Differential Geom. <strong>18</strong> 3 (1983) 523-557 [<a href="https://projecteuclid.org/journals/journal-of-differential-geometry/volume-18/issue-3/The-local-structure-of-Poisson-manifolds/10.4310/jdg/1214437787.full">doi:10.4310/jdg/1214437787</a>]</li> </ul> <p>Monograph on <a class="existingWikiWord" href="/nlab/show/Poisson+geometry">Poisson geometry</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marius+Crainic">Marius Crainic</a>, <a class="existingWikiWord" href="/nlab/show/Rui+Loja+Fernandes">Rui Loja Fernandes</a>, <em>Lectures on Poisson Geometry</em>, Graduate Studies in Mathematics <strong>217</strong>, Amer. Math. Soc. (2021) [<a href="https://bookstore.ams.org/view?ProductCode=GSM/217">ISBN:978-1-4704-6430-1</a>]</li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Poisson_manifold">Poisson manifold</a></em></li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> includes:</p> <ul> <li id="BeniniSchenkel16"><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <em>Poisson algebras for non-linear field theories in the Cahiers topos</em>, Annales Henri Poincar'e, 18(4), 1435-1464 (2017) (<a href="https://arxiv.org/abs/1602.00708">arXiv:1602.00708</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 4, 2023 at 09:33:03. 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