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class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#ForLieAlgebras'>For Lie algebras</a></li> <li><a href='#for_algebras'>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras</a></li> </ul> <li><a href='#existence'>Existence</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#IsomorphismProblem'>Isomorphism problem</a></li> <li><a href='#poisson_algebra_structure_on_'>Poisson algebra structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathfrak{g})</annotation></semantics></math></a></li> <li><a href='#hopf_algebra_structure_on_'>Hopf algebra structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathfrak{g})</annotation></semantics></math></a></li> <li><a href='#pbw_theorem'>PBW theorem</a></li> <li><a href='#relation_to_formal_deformation_quantization'>Relation to formal deformation quantization</a></li> <li><a href='#RelationToTheGroupAlgebra'>Relation to the group algebra</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <h3 id="ForLieAlgebras">For Lie algebras</h3> <p>Given a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+object">Lie algebra object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to some <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = (C,\otimes, \mathbf{1},\tau)</annotation></semantics></math>, an <strong>enveloping monoid</strong> (or <strong>enveloping algebra</strong>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>Lie</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \colon L\to Lie(A)</annotation></semantics></math> of Lie algebras in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> where <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> is a monoid (= algebra) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lie</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Lie(A)</annotation></semantics></math> is the underlying object 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> equipped with the <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><msub><mo stretchy="false">]</mo> <mrow><mi>Lie</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mi>μ</mi><mo>−</mo><mi>μ</mi><mo>∘</mo><msub><mi>τ</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[,]_{Lie(A)}=\mu-\mu\circ\tau_{A,A}</annotation></semantics></math>.</p> <p>A morphism of enveloping algebras</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>Lie</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>⟶</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>A</mi><mo>′</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; \big(A, f \colon L\to Lie(A)\big) \longrightarrow \big(A, f' \colon L\to A'\big) </annotation></semantics></math></div> <p>is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \colon A\to A'</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/monoid+objects">monoid objects</a> completing a commutative triangle of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g\circ f = f'</annotation></semantics></math>. With an obvious composition of morphisms, the enveloping algebras of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <p>A <strong>universal enveloping algebra</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is any universal <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>L</mi></msub><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i_L \colon L\to U(L)</annotation></semantics></math> in the category of enveloping algebras of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>, which implies that it is unique up to an isomorphism if it exists.</p> <p>If it exists for all Lie algebras in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, then the rule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>↦</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L\mapsto U(L)</annotation></semantics></math> can be extended to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> which is the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lie</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>↦</mo><mi>Lie</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Lie \colon A\mapsto Lie(A)</annotation></semantics></math> defined above and the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>L</mi></msub><mo>:</mo><mi>L</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i_L:L\to U(L)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit of the adjunction</a>.</p> <p> <div class='num_remark' id='ForSuperLieAlgebras'> <h6>Remark</h6> <p><strong>(for <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a> as eg. in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> generality of the above definitions is used notably in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> where the relevant Lie algebras arise from <a class="existingWikiWord" href="/nlab/show/Whitehead+brackets">Whitehead brackets</a> which are <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a> (see <a href="super+Lie+algebra#AsLieAlgebrasInternalToSuperVectorSpaces">there</a>), i.e. <a class="existingWikiWord" href="/nlab/show/Lie+algebra+objects">Lie algebra objects</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> (cf. eg. <a href="#MilnorMoore65">Milnor & Moore 1965, §5</a>; <a href="#MayPonto12">May & Ponto 2012, Def. 22.1.3</a>).</p> </div> </p> <h3 id="for_algebras">For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras</h3> <p>In the more general context of <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> there is a notion of universal enveloping <a class="existingWikiWord" href="/nlab/show/E-n+algebra">E-n algebra</a> of an <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra">L-infinity algebra</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> which generalizes the notion of universal associative algebra envelope of a Lie algebra. See at <em><a class="existingWikiWord" href="/nlab/show/universal+enveloping+E-n+algebra">universal enveloping E-n algebra</a></em>.</p> <h2 id="existence">Existence</h2> <p>The existence of the universal enveloping algebra is easy in many concrete symmetric monoidal categories, e.g. the symmetric monoidal category of bounded chain complexes (giving the universal enveloping <a class="existingWikiWord" href="/nlab/show/differential+graded+algebra">dg-algebra</a> of a <a class="existingWikiWord" href="/nlab/show/differential+graded+Lie+algebra">dg-Lie algebra</a>), but not true in general.</p> <p>First of all if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> admits countable coproducts, form the tensor algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TL</mi><mo>=</mo><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow> <mn>∞</mn></msubsup><msup><mi>L</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">TL=\coprod_{n=0}^\infty L^{\otimes n}</annotation></semantics></math> on the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>; this is a monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. In most standard cases, one can also form the smallest 2-sided ideal (i.e. <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>-subbimodule) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> in monoid <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> among those ideals whose inclusion into <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> is factorizing the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mo>,</mo><mo stretchy="false">]</mo><mo>−</mo><msub><mi>m</mi> <mi>TL</mi></msub><mo>+</mo><msub><mi>m</mi> <mi>TL</mi></msub><mo>∘</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>∘</mo><mo>⊗</mo><mo>:</mo><mi>L</mi><mo>⊗</mo><mi>L</mi><mo>→</mo><mi>TL</mi></mrow><annotation encoding="application/x-tex">([,]-m_{TL}+m_{TL}\circ\tau)\circ \otimes :L\otimes L\to TL</annotation></semantics></math>; if the coequalizers exist in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> then we can form the quotient object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TL</mi><mo stretchy="false">/</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">TL/I</annotation></semantics></math> and there is an induced monoid structure in it. Under mild conditions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, the natural morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>L</mi></msub><mo>:</mo><mi>L</mi><mo>→</mo><mi>TL</mi><mo stretchy="false">/</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i_L:L\to TL/I</annotation></semantics></math> is an universal enveloping monoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an abelian tensor category and some flatness conditions on the tensor product are satisfied, then the enveloping monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>L</mi></msub><mo>:</mo><mi>L</mi><mo>→</mo><mi>TL</mi><mo stretchy="false">/</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i_L:L\to TL/I</annotation></semantics></math> is a monic morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>L</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(L\coprod L)\cong U(L)\otimes U(L)</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="IsomorphismProblem">Isomorphism problem</h3> <p>Notice that the enveloping algebras of two Lie algebras may be <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> as <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a> even if the two Lie algebras they arise from are not isomorphic to each other.</p> <p>This happens even in <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>, in which case, however, at least the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of the Lie algebra is encoded in its universal enveloping algebra, in the guise of the <em>Gelfand-Kirillov dimension</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GK</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">GK\big(U(L)\big)</annotation></semantics></math>.</p> <h3 id="poisson_algebra_structure_on_">Poisson algebra structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathfrak{g})</annotation></semantics></math></h3> <p>The universal enveloping algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathfrak{g})</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> is naturally a (non-commutative) <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a> with the restriction of the Poisson bracket to generators being the original <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a></p> <h3 id="hopf_algebra_structure_on_">Hopf algebra structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathfrak{g})</annotation></semantics></math></h3> <p>Suppose the universal enveloping algebras of Lie algebras exist in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-linear symmetric monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and the functorial choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>↦</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L\mapsto U(L)</annotation></semantics></math> realizing the above construction with tensor products is fixed. For example, this is true in the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>. Then the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">L\to 0</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> is the trivial Lie algebra) induces the counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\epsilon:U(L)\to U(0)=\mathbf{1}</annotation></semantics></math>. The coproduct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo>×</mo><mi>L</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta:U(L)\to U(L\times L)\cong U(L)\otimes U(L)</annotation></semantics></math> is induced by the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>→</mo><mi>L</mi><mo>×</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">L\to L\times L</annotation></semantics></math> whereas the antipode <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>id</mi><mo stretchy="false">)</mo><mo>:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S=U(-id):U(L)\to U(L)</annotation></semantics></math>. One checks that these morphisms make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(L)</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. (e.g <a href="#MilnorMoore65">Milnor-Moore 65, section 5</a>) The <em><a class="existingWikiWord" href="/nlab/show/Milnor-Moore+theorem">Milnor-Moore theorem</a></em> states conditions under which the converse holds (hence under which a primitively generated Hopf algebra is a universal enveloping algebra of a Lie algebra).</p> <p>If the category is simply the vector spaces over a field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">l\in L</annotation></semantics></math>, after we identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> with its image in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(L)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mo>⊗</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>⊗</mo><mi>l</mi></mrow><annotation encoding="application/x-tex">\Delta(l) = l\otimes 1 + 1\otimes l</annotation></semantics></math>, i.e. the elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/primitive+element">primitive element</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(L)</annotation></semantics></math>.</p> <h3 id="pbw_theorem">PBW theorem</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt+theorem">Poincaré–Birkhoff–Witt theorem</a> states that the <a class="existingWikiWord" href="/nlab/show/associated+graded">associated graded</a> algebra of an enveloping algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(g)</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a> is canonically isomorphic to a <a class="existingWikiWord" href="/nlab/show/symmetric+algebra">symmetric algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sym</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sym(g)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(g)</annotation></semantics></math> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(g)</annotation></semantics></math> as a coalgebra, via the projection map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Gr</mi><mi>U</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(g)\to Gr U(g)</annotation></semantics></math>.</p> <h3 id="relation_to_formal_deformation_quantization">Relation to formal deformation quantization</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a></em> the section <em><a href="formal+deformation+quantization#RelationToUniversalEnvelopingAlgebras">Relation to universal enveloping algebras</a></em>.</p> <h3 id="RelationToTheGroupAlgebra">Relation to the group algebra</h3> <blockquote> <p>The universal enveloping algebra of a Lie algebra is the analogue of the usual <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a> of a group. It has the analogous function of exhibiting the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/Lie+algebra+modules">Lie algebra modules</a> as a category of <a class="existingWikiWord" href="/nlab/show/modules">modules</a> for an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>. This becomes more than an analogy when the universal enveloping algebra is viewed with its full <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> structure. By dualization, one obtains a commutative Hopf algebra which, in the case where the Lie algebra is that of an irreducible <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+0">characteristic 0</a>, contains the algebra of polynomial functions of that group as a sub Hopf algebra in a natural fashion.</p> </blockquote> <p>(quoted from <a href="#Hochschild81">Hochschild 1981, p. 221</a>, see <a href="#Hochschild81">ibid. Thm. 3.1 on p. 230</a>; <a href="#Tjin92">Tjin 1992, Thm. 1</a>)</p> <h2 id="examples">Examples</h2> <p> <div class='num_remark' id='WeylAlgebraAndHeisenbergAlgebra'> <h6>Example</h6> <p>Consider the standard <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> on 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> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2n}</annotation></semantics></math>, making a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a>. This gives rise to the corresponding <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a>.</p> <p>Depending on conventions, the <a class="existingWikiWord" href="/nlab/show/universal+enveloping+algebra">universal enveloping algebra</a> of the <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a> either already is the <a class="existingWikiWord" href="/nlab/show/Weyl+algebra">Weyl algebra</a> on <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> generators or else it becomes so after after forming the <a class="existingWikiWord" href="/nlab/show/quotient+algebra">quotient algebra</a> in which the central generator is identified with the <a class="existingWikiWord" href="/nlab/show/unit+element">unit element</a> of the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> – whereas in the former case (considered eg. in <a href="#Kravchenko00">Kravchenko 2000, Def. 2.1</a>; <a href="#Bekaert05">Bekaert 2005, p. 9</a>) the central generator plays the role of the formal <a class="existingWikiWord" href="/nlab/show/Planck+constant">Planck constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math> with the Weyl algebra regarded as a <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a> of the <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><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2m}</annotation></semantics></math>.</p> <p>Accordingly, given a <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+n-algebra">Heisenberg Lie <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-algebra</a> it makes sense to call its <a class="existingWikiWord" href="/nlab/show/universal+enveloping+E-n+algebra">universal enveloping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>E</mi> <mi>n</mi></msub> </mrow> <annotation encoding="application/x-tex">E_n</annotation> </semantics> </math>-algebra</a> a <em><a class="existingWikiWord" href="/nlab/show/Weyl+n-algebra">Weyl <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-algebra</a></em>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p><strong>(Universal enveloping of a tangent Lie algebra)</strong> <br /> The universal enveloping algebra of the <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebra">tangent Lie algebra</a> of a finite-dimensional <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> over the real or complex numbers is canonically isomorphic to the algebra of the left invariant <a class="existingWikiWord" href="/nlab/show/differential+operators">differential operators</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+series">Hausdorff series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt+theorem">Poincaré–Birkhoff–Witt theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coexponential+map">coexponential map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Duflo+isomorphism">Duflo isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/oidification">oidification</a> to <a class="existingWikiWord" href="/nlab/show/universal+enveloping+algebroid">universal enveloping algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+enveloping+E-n+algebra">universal enveloping E-n algebra</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="Serre64"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Serre">Jean-Pierre Serre</a>: <em>Universal Algebra of a Lie Algebra</em>, Chapter III in: <em>Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University</em>, Lecture Notes in Mathematics <strong>1500</strong>, Springer (1992) [<a href="https://doi.org/10.1007/978-3-540-70634-2">doi:10.1007/978-3-540-70634-2</a>]</p> </li> <li id="Dixmier74"> <p><a class="existingWikiWord" href="/nlab/show/Jacques+Dixmier">Jacques Dixmier</a>; <em><a class="existingWikiWord" href="/nlab/show/Alg%C3%A8bres+Enveloppantes">Algèbres Enveloppantes</a></em>, Cahiers Scientifique (1974), Engl transl: <em>Enveloping Algebras</em>, Graduate Studies in Mathematics <strong>11</strong> American Mathematical Society (1996) [<a href="https://bookstore.ams.org/gsm-11">ams:gsm-11</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, §I.2 in: <em>Lie groups and Lie algebras – Chapters 1-3</em>, Springer (1975, 1989) [<a href="https://link.springer.com/book/9783540642428">ISBN:9783540642428</a>]</p> </li> <li id="Hochschild81"> <p><a class="existingWikiWord" href="/nlab/show/Gerhard+P.+Hochschild">Gerhard P. Hochschild</a>, <em>The Universal Enveloping Algebra</em>, Chapter XVI in: <em>Basic Theory of Algebraic Groups and Lie Algebras</em>, Graduate Texts in Mathematics <strong>75</strong>, Springer (1981) [<a href="https://doi.org/10.1007/978-1-4613-8114-3_16">doi:10.1007/978-1-4613-8114-3_16</a>]</p> </li> </ul> <p>Further discussion in the context of <a class="existingWikiWord" href="/nlab/show/quantum+groups">quantum groups</a>:</p> <ul> <li id="Tjin92"><a class="existingWikiWord" href="/nlab/show/Tjark+Tjin">Tjark Tjin</a>, <em>An introduction to quantized Lie groups and algebras</em>, Int. J. Mod. Phys. A <strong>7</strong> (1992) 6175-6213 [<a href="https://arxiv.org/abs/hep-th/9111043">arXiv:hep-th/9111043</a>, <a href="https://doi.org/10.1142/S0217751X92002805">doi:10.1142/S0217751X92002805</a>]</li> </ul> <p>Discussion in the generality of <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a> such as notably of the <a class="existingWikiWord" href="/nlab/show/Whitehead+Lie+algebras">Whitehead Lie algebras</a> arising in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>:</p> <ul> <li id="MilnorMoore65"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <a class="existingWikiWord" href="/nlab/show/John+Moore">John Moore</a>, §5 in <em>On the structure of Hopf algebras</em>, Annals of Math. <strong>81</strong> (1965) 211-264 [<a href="https://doi.org/10.2307/1970615">doi:10.2307/1970615</a>, <a href="http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf">pdf</a>]</p> </li> <li id="MayPonto12"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Kate+Ponto">Kate Ponto</a>, Def. 22.1.3 in: <em>More concise algebraic topology – Localization, Completion, and Model Categories</em>, University of Chicago Press (2012) [<a href="https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html">ISBN:9780226511795</a>, <a href="https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf">pdf</a>]</p> </li> </ul> <p>See also:</p> <ul> <li> <p>Wikipedia: <em><a href="http://en.wikipedia.org/wiki/Universal_enveloping_algebra">Universal enveloping algebra</a></em>, <em><a href="http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem">PBW theorem</a></em></p> </li> <li> <p>F. A. Berezin, <em>Some remarks about the associated envelope of a Lie algebra</em>, Func. Analysis and Its Appl. 1967, 1:2, 91–102; Rus. original in Fun. Anal. Pril. 1:2 (1967) 1–14 <a href="http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=2813&volume=1&year=1967&issue=2&fpage=1&what=fullt&option_lang=eng">Rus. pdf</a> <a href="http://www.ams.org/mathscinet-getitem?mr=219671">MR219671</a></p> </li> <li> <p>MathOverflow question <a href="http://mathoverflow.net/questions/25020/what-is-the-universal-enveloping-algebra">What is the universal enveloping algebra</a> which is looking for a rather general construction in a class of symmetric monoidal <a class="existingWikiWord" href="/nlab/show/pseudoabelian+category">pseudoabelian categories</a>.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 5, 2025 at 01:25:59. 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