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topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+arithmetic+geometry">higher arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+arithmetic+geometry">E-∞ arithmetic geometry</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/number">number</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, <a class="existingWikiWord" href="/nlab/show/integer+number">integer number</a>, <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>, <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, <a class="existingWikiWord" href="/nlab/show/irrational+number">irrational number</a>, <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a>, <a class="existingWikiWord" href="/nlab/show/quaternion">quaternion</a>, <a class="existingWikiWord" href="/nlab/show/octonion">octonion</a>, <a class="existingWikiWord" href="/nlab/show/adic+number">adic number</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+number">cardinal number</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+number">ordinal number</a>, <a class="existingWikiWord" href="/nlab/show/surreal+number">surreal number</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+arithmetic">Peano arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/second-order+arithmetic">second-order arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transfinite+arithmetic">transfinite arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+arithmetic">cardinal arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+arithmetic">ordinal arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+field">prime field</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+integer">p-adic integer</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+rational+number">p-adic rational number</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+complex+number">p-adic complex number</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></strong>, <a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+scheme">arithmetic scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+curve">arithmetic curve</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chern-Simons+theory">arithmetic Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chow+group">arithmetic Chow group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil-%C3%A9tale+topology+for+arithmetic+schemes">Weil-étale topology for arithmetic schemes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absolute+cohomology">absolute cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+conjecture+on+Tamagawa+numbers">Weil conjecture on Tamagawa numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borger%27s+absolute+geometry">Borger's absolute geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Iwasawa-Tate+theory">Iwasawa-Tate theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adelic+integration">adelic integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shtuka">shtuka</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenioid">Frobenioid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Arakelov+geometry">Arakelov geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Riemann-Roch+theorem">arithmetic Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+algebraic+K-theory">differential algebraic K-theory</a></p> </li> </ul> </div></div> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#normed_division_algebra_structure'>Normed division algebra structure</a></li> <li><a href='#modules_and_bimodules'>Modules and bimodules</a></li> <li><a href='#automorphisms'>Automorphisms</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>A <em>quaternion</em> or <em>Hamilton number</em> is a kind of <a class="existingWikiWord" href="/nlab/show/number">number</a> similar to the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> but with three instead of one <a class="existingWikiWord" href="/nlab/show/square+root">square root</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1)</annotation></semantics></math> adjoined, satisfying certain relations.</p> <p>The <strong>quaternions</strong> form the largest <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative</a> <a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a>, usually denoted <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">\mathbb{H}</annotation></semantics></math> after <a class="existingWikiWord" href="/nlab/show/William+Rowan+Hamilton">William Rowan Hamilton</a> (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> is taken for the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>).</p> <h2 id="properties">Properties</h2> <h3 id="normed_division_algebra_structure">Normed division algebra structure</h3> <p>The structure of <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">\mathbb{H}</annotation></semantics></math> as an <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/associative+unital+algebra">algebra</a> is given by a basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{1, i, j, k\}</annotation></semantics></math> of the underlying <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> of <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">\mathbb{H}</annotation></semantics></math>, equipped with a multiplication table where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/identity+element">identity element</a> and otherwise uniquely specified by the equations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>j</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>k</mi> <mn>2</mn></msup><mo>=</mo><mi>i</mi><mi>j</mi><mi>k</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo></mrow><annotation encoding="application/x-tex">i^2 = j^2 = k^2 = i j k = -1,</annotation></semantics></math></div> <p>and extended by <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>-linearity to all of <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">\mathbb{H}</annotation></semantics></math>. The norm on <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">\mathbb{H}</annotation></semantics></math> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow><mo stretchy="false">‖</mo><mi>α</mi><mo stretchy="false">‖</mo></mrow> <mn>2</mn></msup><mo>=</mo><mi>α</mi><mover><mi>α</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">{\|\alpha\|}^2 = \alpha \widebar{\alpha}</annotation></semantics></math></div> <p>where given an <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+combination">linear combination</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>=</mo><mi>a</mi><mn>1</mn><mo>+</mo><mi>b</mi><mi>i</mi><mo>+</mo><mi>c</mi><mi>j</mi><mo>+</mo><mi>d</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">\alpha = a 1 + b i + c j + d k</annotation></semantics></math>, we define the <em>conjugate</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>α</mi><mo>¯</mo></mover><mo>≔</mo><mi>a</mi><mn>1</mn><mo>−</mo><mi>b</mi><mi>i</mi><mo>−</mo><mi>c</mi><mi>j</mi><mo>−</mo><mi>d</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">\widebar{\alpha} \coloneqq a 1 - b i - c j - d k</annotation></semantics></math>. A simple calculation yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow><mo stretchy="false">‖</mo><mi>α</mi><mo stretchy="false">‖</mo></mrow> <mn>2</mn></msup><mo>=</mo><msup><mi>a</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>b</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>d</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">{\|\alpha\|}^2 = a^2 + b^2 + c^2 + d^2</annotation></semantics></math></div> <p>whence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\alpha \neq 0</annotation></semantics></math>, the multiplicative <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>α</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">‖</mo><mi>α</mi><mo stretchy="false">‖</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mover><mi>α</mi><mo>¯</mo></mover><mo>.</mo></mrow><annotation encoding="application/x-tex">\alpha^{-1} = \frac1{{\|\alpha\|}^2} \widebar{\alpha}.</annotation></semantics></math></div> <p>In this way <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">\mathbb{H}</annotation></semantics></math> is a normed division algebra.</p> <h3 id="modules_and_bimodules">Modules and bimodules</h3> <p>We have canonical left and right <a class="existingWikiWord" href="/nlab/show/module">module</a> structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℍ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{H}^n</annotation></semantics></math>, but as <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">\mathbb{H}</annotation></semantics></math> is not commutative, if we want to talk about tensor products of modules, we need to consider <a class="existingWikiWord" href="/nlab/show/bimodules">bimodules</a>. This also means that ordinary <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a> as is used over a field is not quite the same when dealing with quaternions. For instance, one needs to distinguish between <em>left</em> and <em>right</em> <a class="existingWikiWord" href="/nlab/show/eigenvalues">eigenvalues</a> of <a class="existingWikiWord" href="/nlab/show/matrices">matrices</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_n(\mathbb{H})</annotation></semantics></math> (using the left and right module structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℍ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{H}^n</annotation></semantics></math> respectively), and only left eigenvalues relate to the <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> of the associated linear operator.</p> <p>Using the conjugation operation one can define an inner product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo stretchy="false">⟩</mo><mo>:</mo><mo>=</mo><mover><mi>q</mi><mo>¯</mo></mover><mi>p</mi></mrow><annotation encoding="application/x-tex">\langle q,p\rangle := \overline{q} p</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℍ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{H}^n</annotation></semantics></math> so that the corresponding <a class="existingWikiWord" href="/nlab/show/orthogonal+group+of+an+inner+product+space">orthogonal group</a> is the <a class="existingWikiWord" href="/nlab/show/compact+symplectic+group">compact symplectic group</a>.</p> <h3 id="automorphisms">Automorphisms</h3> <div class="num_prop" id="AutomorphismsOfQUatrnionsAlgebraIsSO3"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of the quaternions, as a real algebra, is <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">SO(3)</a>, <a class="existingWikiWord" href="/nlab/show/action">acting</a> canonically on their imaginary part (in generalization of how the product of <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> respects the <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a> action)</p> </div> <p>See also at <em><a href="normed+division+algebra#Automorphisms">normed division algebra – automorphism</a></em></p> <p>(e.g. <a href="#KlimovZhuravlev">Klimov-Zhuravlev, p. 85</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternionic+vector+space">quaternionic vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternionic+vector+bundle">quaternionic vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dieudonn%C3%A9+determinant">Dieudonné determinant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/SL%282%2CH%29">SL(2,H)</a></p> </li> </ul> <div> <p><strong><a href="spin+group#ExceptionalIsomorphisms">exceptional</a> <a class="existingWikiWord" href="/nlab/show/spin+representation">spinors</a> and <a class="existingWikiWord" href="/nlab/show/real+numbers">real</a> <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Lorentzian+spacetime">Lorentzian</a> <br /> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <br /> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> entry</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><mn>3</mn><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3 = 2+1</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><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2,1) \simeq SL(2,\mathbb{R})</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>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">super 1-brane in 3d</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4 = 3+1</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><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3,1) \simeq SL(2, \mathbb{C})</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>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+2-brane+in+4d">super 2-brane in 4d</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>6</mn><mo>=</mo><mn>5</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">6 = 5+1</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><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">Spin(5,1) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SL%282%2CH%29">SL(2,H)</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> <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">\mathbb{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string">little string</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>10</mn><mo>=</mo><mn>9</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">10 = 9+1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%2C1%29">Spin(9,1)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">{\simeq}</annotation></semantics></math> “<a class="existingWikiWord" href="/nlab/show/SL%282%2CO%29">SL(2,O)</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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕆</mi></mrow><annotation encoding="application/x-tex">\mathbb{O}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic</a>/<a class="existingWikiWord" href="/nlab/show/type+II+string">type II string</a></td></tr> </tbody></table> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry+and+division+algebra">supersymmetry and division algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternionic+quantum+mechanics">quaternionic quantum mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternion+group">quaternion group</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+unitary+group">quaternionic unitary group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tessarines">tessarines</a></p> </li> </ul> <h2 id="references">References</h2> <p>Monographs:</p> <ul> <li id="KlimovZhuravlev"> <p>D. M. Klimov, V. Ph. Zhuravlev, <em>Group-Theoretic Methods in Mechanics and Applied Mathematics</em>, Routledge (2004, 2020) [<a href="https://www.routledge.com/Group-Theoretic-Methods-in-Mechanics-and-Applied-Mathematics/Klimov-Zhuravlev/p/book/9780367446987">ISBN:9780367446987</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ernst+Binz">Ernst Binz</a>, Sonja Pods, Ch 1 in: <em>The geometry of Heisenberg groups — With Applications in Signal Theory, Optics, Quantization, and Field Quantization</em>, Mathematical Surveys and Monographs <strong>151</strong>, American Mathematical Society (2008) [<a href="https://bookstore.ams.org/surv-151">ams:surv-151</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tevian+Dray">Tevian Dray</a>, <a class="existingWikiWord" href="/nlab/show/Corinne+Manogue">Corinne Manogue</a>, Section 3.1 of: <em>The Geometry of Octonions</em>, World Scientific (2015) [<a href="https://doi.org/10.1142/8456">doi:10.1142/8456</a>]</p> </li> </ul> <p>See also:</p> <ul> <li>T. Y. Lam, <em>Hamilton’s Quaternions</em> (<a href="http://math.berkeley.edu/~lam/quat.ps">ps</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 21, 2024 at 02:07:18. 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