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class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Supercommutativity</span> </div> </a> <ul id="toc-Supercommutativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Super_tensor_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Super_tensor_product"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Super tensor product</span> </div> </a> <ul id="toc-Super_tensor_product-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations_and_categorical_definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations_and_categorical_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Generalizations and categorical definition</span> </div> </a> <ul id="toc-Generalizations_and_categorical_definition-sublist" class="vector-toc-list"> </ul> </li> <li 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algebraic structure used in theoretical physics</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>, a <b>superalgebra</b> is a <b>Z</b><sub>2</sub>-<a href="/wiki/Graded_algebra" class="mw-redirect" title="Graded algebra">graded algebra</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> That is, it is an <a href="/wiki/Algebra_(ring_theory)" class="mw-redirect" title="Algebra (ring theory)">algebra</a> over a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> or <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. </p><p>The prefix <i>super-</i> comes from the theory of <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a> in theoretical physics. Superalgebras and their representations, <a href="/wiki/Supermodule" title="Supermodule">supermodules</a>, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called <a href="/wiki/Super_linear_algebra" class="mw-redirect" title="Super linear algebra">super linear algebra</a>. Superalgebras also play an important role in related field of <a href="/wiki/Supergeometry" title="Supergeometry">supergeometry</a> where they enter into the definitions of <a href="/wiki/Graded_manifold" title="Graded manifold">graded manifolds</a>, <a href="/wiki/Supermanifold" title="Supermanifold">supermanifolds</a> and <a href="/w/index.php?title=Superscheme&action=edit&redlink=1" class="new" title="Superscheme (page does not exist)">superschemes</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=1" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>K</i> be a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>. In most applications, <i>K</i> is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 0, such as <b>R</b> or <b>C</b>. </p><p>A <b>superalgebra</b> over <i>K</i> is a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)"><i>K</i>-module</a> <i>A</i> with a <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> decomposition </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=A_{0}\oplus A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=A_{0}\oplus A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434041496b1aa461b29c64dea1c9bd6461ae4def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.277ex; height:2.509ex;" alt="{\displaystyle A=A_{0}\oplus A_{1}}"></span></dd></dl> <p>together with a <a href="/wiki/Bilinear_map" title="Bilinear map">bilinear</a> multiplication <i>A</i> × <i>A</i> → <i>A</i> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}A_{j}\subseteq A_{i+j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⊆</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}A_{j}\subseteq A_{i+j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c90600a5635b57143e212ab5b3cf6968501526a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.793ex; height:2.843ex;" alt="{\displaystyle A_{i}A_{j}\subseteq A_{i+j}}"></span></dd></dl> <p>where the subscripts are read <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> 2, i.e. they are thought of as elements of <b>Z</b><sub>2</sub>. </p><p>A <b>superring</b>, or <b>Z</b><sub>2</sub>-<a href="/wiki/Graded_ring" title="Graded ring">graded ring</a>, is a superalgebra over the ring of <a href="/wiki/Integer" title="Integer">integers</a> <b>Z</b>. </p><p>The elements of each of the <i>A</i><sub><i>i</i></sub> are said to be <b>homogeneous</b>. The <b>parity</b> of a homogeneous element <i>x</i>, denoted by |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>|, is 0 or 1 according to whether it is in <i>A</i><sub>0</sub> or <i>A</i><sub>1</sub>. Elements of parity 0 are said to be <b>even</b> and those of parity 1 to be <b>odd</b>. If <i>x</i> and <i>y</i> are both homogeneous then so is the product <i>xy</i> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |xy|=|x|+|y|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |xy|=|x|+|y|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e88e5dfc89eabad4e4915ffd568a3a9a99ec07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.79ex; height:2.843ex;" alt="{\displaystyle |xy|=|x|+|y|}"></span>. </p><p>An <b>associative superalgebra</b> is one whose multiplication is <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a> and a <b>unital superalgebra</b> is one with a multiplicative <a href="/wiki/Identity_element" title="Identity element">identity element</a>. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital. </p><p>A <b><a href="/wiki/Commutative_superalgebra" class="mw-redirect" title="Commutative superalgebra">commutative superalgebra</a></b> (or supercommutative algebra) is one which satisfies a graded version of <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a>. Specifically, <i>A</i> is commutative if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle yx=(-1)^{|x||y|}xy\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mi>x</mi> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle yx=(-1)^{|x||y|}xy\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8445b7870c288264a57a05d0005488666d11a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.055ex; height:3.343ex;" alt="{\displaystyle yx=(-1)^{|x||y|}xy\,}"></span></dd></dl> <p>for all homogeneous elements <i>x</i> and <i>y</i> of <i>A</i>. There are superalgebras that are commutative in the ordinary sense, but not in the superalgebra sense. For this reason, commutative superalgebras are often called <i>supercommutative</i> in order to avoid confusion.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Sign_conventions">Sign conventions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=2" title="Edit section: Sign conventions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When the <b>Z</b><sub>2</sub> grading arises as a "rollup" of a <b>Z</b>- or <b>N</b>-<a href="/wiki/Graded_algebra" class="mw-redirect" title="Graded algebra">graded algebra</a> into even and odd components, then two distinct (but essentially equivalent) sign conventions can be found in the literature.<sup id="cite_ref-deligne_3-0" class="reference"><a href="#cite_note-deligne-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> These can be called the "cohomological sign convention" and the "super sign convention". They differ in how the antipode (exchange of two elements) behaves. In the first case, one has an exchange map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\mapsto (-1)^{mn+pq}yx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> <mo>+</mo> <mi>p</mi> <mi>q</mi> </mrow> </msup> <mi>y</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\mapsto (-1)^{mn+pq}yx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c735f845e4b619e878147347a0d64b5818fd81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.887ex; height:3.009ex;" alt="{\displaystyle xy\mapsto (-1)^{mn+pq}yx}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=\deg x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>deg</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=\deg x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69285520a6a64ff4715594143e1d37dbd6db3ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.343ex; height:2.509ex;" alt="{\displaystyle m=\deg x}"></span> is the degree (<b>Z</b>- or <b>N</b>-grading) of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> the parity. Likewise, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\deg y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>deg</mi> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\deg y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0779cd82590203a501969954bd5fae716fb49e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.523ex; height:2.509ex;" alt="{\displaystyle n=\deg y}"></span> is the degree of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and with parity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b77c4dfff8774d73f815f799aa68d83a96d7095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.716ex; height:2.009ex;" alt="{\displaystyle q.}"></span> This convention is commonly seen in conventional mathematical settings, such as differential geometry and differential topology. The other convention is to take </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\mapsto (-1)^{pq}yx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>q</mi> </mrow> </msup> <mi>y</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\mapsto (-1)^{pq}yx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6cc5e102ccf787d27a8832be188e4bcba4a8df3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.18ex; height:2.843ex;" alt="{\displaystyle xy\mapsto (-1)^{pq}yx}"></span></dd></dl> <p>with the parities given as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=m{\bmod {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=m{\bmod {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc06244127722744fca1f451529263941ecb6a5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.241ex; height:2.509ex;" alt="{\displaystyle p=m{\bmod {2}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=n{\bmod {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=n{\bmod {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9518ed9e0778d62e76e95dd788fa918a21209a3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.406ex; height:2.509ex;" alt="{\displaystyle q=n{\bmod {2}}}"></span> the parity. This is more often seen in physics texts, and requires a parity functor to be judiciously employed to track isomorphisms. Detailed arguments are provided by <a href="/wiki/Pierre_Deligne" title="Pierre Deligne">Pierre Deligne</a><sup id="cite_ref-deligne_3-1" class="reference"><a href="#cite_note-deligne-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Any algebra over a commutative ring <i>K</i> may be regarded as a purely even superalgebra over <i>K</i>; that is, by taking <i>A</i><sub>1</sub> to be trivial.</li> <li>Any <b>Z</b>- or <b>N</b>-<a href="/wiki/Graded_algebra" class="mw-redirect" title="Graded algebra">graded algebra</a> may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as <a href="/wiki/Tensor_algebra" title="Tensor algebra">tensor algebras</a> and <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial rings</a> over <i>K</i>.</li> <li>In particular, any <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> over <i>K</i> is a superalgebra. The exterior algebra is the standard example of a <a href="/wiki/Supercommutative_algebra" title="Supercommutative algebra">supercommutative algebra</a>.</li> <li>The <a href="/wiki/Symmetric_polynomials" class="mw-redirect" title="Symmetric polynomials">symmetric polynomials</a> and <a href="/wiki/Alternating_polynomials" class="mw-redirect" title="Alternating polynomials">alternating polynomials</a> together form a superalgebra, being the even and odd parts, respectively. Note that this is a different grading from the grading by degree.</li> <li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a> are superalgebras. They are generally noncommutative.</li> <li>The set of all <a href="/wiki/Endomorphism" title="Endomorphism">endomorphisms</a> (denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {End} (V)\equiv \mathbf {Hom} (V,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> <mi mathvariant="bold">n</mi> <mi mathvariant="bold">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> <mi mathvariant="bold">o</mi> <mi mathvariant="bold">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {End} (V)\equiv \mathbf {Hom} (V,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5563ea830332c02e25fee41589b9130416a79bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.495ex; height:2.843ex;" alt="{\displaystyle \mathbf {End} (V)\equiv \mathbf {Hom} (V,V)}"></span>, where the boldface <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea3aafc91ebbd147d45c3c69e88431c48cbe9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle \mathrm {Hom} }"></span> is referred to as <i>internal</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea3aafc91ebbd147d45c3c69e88431c48cbe9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle \mathrm {Hom} }"></span>, composed of <i>all</i> linear maps) of a <a href="/wiki/Super_vector_space" title="Super vector space">super vector space</a> forms a superalgebra under composition.</li> <li>The set of all square <a href="/wiki/Supermatrices" class="mw-redirect" title="Supermatrices">supermatrices</a> with entries in <i>K</i> forms a superalgebra denoted by <i>M</i><sub><i>p</i>|<i>q</i></sub>(<i>K</i>). This algebra may be identified with the algebra of endomorphisms of a free supermodule over <i>K</i> of rank <i>p</i>|<i>q</i> and is the internal Hom of above for this space.</li> <li><a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebras</a> are a graded analog of <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a>. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a <a href="/wiki/Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a> of a Lie superalgebra which is a unital, associative superalgebra.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_definitions_and_constructions">Further definitions and constructions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=4" title="Edit section: Further definitions and constructions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Even_subalgebra">Even subalgebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=5" title="Edit section: Even subalgebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>A</i> be a superalgebra over a commutative ring <i>K</i>. The <a href="/wiki/Submodule" class="mw-redirect" title="Submodule">submodule</a> <i>A</i><sub>0</sub>, consisting of all even elements, is closed under multiplication and contains the identity of <i>A</i> and therefore forms a <a href="/wiki/Subalgebra" title="Subalgebra">subalgebra</a> of <i>A</i>, naturally called the <b>even subalgebra</b>. It forms an ordinary <a href="/wiki/Algebra_(ring_theory)" class="mw-redirect" title="Algebra (ring theory)">algebra</a> over <i>K</i>. </p><p>The set of all odd elements <i>A</i><sub>1</sub> is an <i>A</i><sub>0</sub>-<a href="/wiki/Bimodule" title="Bimodule">bimodule</a> whose scalar multiplication is just multiplication in <i>A</i>. The product in <i>A</i> equips <i>A</i><sub>1</sub> with a <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu :A_{1}\otimes _{A_{0}}A_{1}\to A_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>:</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu :A_{1}\otimes _{A_{0}}A_{1}\to A_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fa93b7373fccbce8bea5ae10a669726b9b1d08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.482ex; height:2.843ex;" alt="{\displaystyle \mu :A_{1}\otimes _{A_{0}}A_{1}\to A_{0}}"></span></dd></dl> <p>such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (x\otimes y)\cdot z=x\cdot \mu (y\otimes z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⊗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>⊗</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (x\otimes y)\cdot z=x\cdot \mu (y\otimes z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bddebf0a13323a2e1dab0d0a0fdd467ca5bc427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.706ex; height:2.843ex;" alt="{\displaystyle \mu (x\otimes y)\cdot z=x\cdot \mu (y\otimes z)}"></span></dd></dl> <p>for all <i>x</i>, <i>y</i>, and <i>z</i> in <i>A</i><sub>1</sub>. This follows from the associativity of the product in <i>A</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Grade_involution">Grade involution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=6" title="Edit section: Grade involution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a canonical <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involutive</a> <a href="/wiki/Automorphism" title="Automorphism">automorphism</a> on any superalgebra called the <b>grade involution</b>. It is given on homogeneous elements by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}=(-1)^{|x|}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}=(-1)^{|x|}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9109d6a37858e20bdd99854f8cf478122753bcf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.625ex; height:3.343ex;" alt="{\displaystyle {\hat {x}}=(-1)^{|x|}x}"></span></dd></dl> <p>and on arbitrary elements by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}=x_{0}-x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}=x_{0}-x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e1f21d78e28f4e36454997c555f6400d8b9e66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.036ex; height:2.509ex;" alt="{\displaystyle {\hat {x}}=x_{0}-x_{1}}"></span></dd></dl> <p>where <i>x</i><sub><i>i</i></sub> are the homogeneous parts of <i>x</i>. If <i>A</i> has no <a href="/wiki/Torsion_(algebra)" title="Torsion (algebra)">2-torsion</a> (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of <i>A</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}=\{x\in A:{\hat {x}}=(-1)^{i}x\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}=\{x\in A:{\hat {x}}=(-1)^{i}x\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8775b6623f713c523bfb6376fc9f2d7a52ed6383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.801ex; height:3.176ex;" alt="{\displaystyle A_{i}=\{x\in A:{\hat {x}}=(-1)^{i}x\}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Supercommutativity">Supercommutativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=7" title="Edit section: Supercommutativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b><a href="/wiki/Supercommutator" class="mw-redirect" title="Supercommutator">supercommutator</a></b> on <i>A</i> is the binary operator given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,y]=xy-(-1)^{|x||y|}yx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mi>y</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,y]=xy-(-1)^{|x||y|}yx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0484e6a70a7eb05b7adebe88fbd8af1a7e0b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.321ex; height:3.343ex;" alt="{\displaystyle [x,y]=xy-(-1)^{|x||y|}yx}"></span></dd></dl> <p>on homogeneous elements, extended to all of <i>A</i> by linearity. Elements <i>x</i> and <i>y</i> of <i>A</i> are said to <b>supercommute</b> if <span class="nowrap">[<i>x</i>, <i>y</i>] = 0</span>. </p><p>The <b>supercenter</b> of <i>A</i> is the set of all elements of <i>A</i> which supercommute with all elements of <i>A</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Z} (A)=\{a\in A:[a,x]=0{\text{ for all }}x\in A\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Z} (A)=\{a\in A:[a,x]=0{\text{ for all }}x\in A\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0553f1ed94b374484e506caab491fac2f339e9d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.839ex; height:2.843ex;" alt="{\displaystyle \mathrm {Z} (A)=\{a\in A:[a,x]=0{\text{ for all }}x\in A\}.}"></span></dd></dl> <p>The supercenter of <i>A</i> is, in general, different than the <a href="/wiki/Center_of_an_algebra" class="mw-redirect" title="Center of an algebra">center</a> of <i>A</i> as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of <i>A</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Super_tensor_product">Super tensor product</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=8" title="Edit section: Super tensor product"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The graded <a href="/wiki/Tensor_product_of_algebras" title="Tensor product of algebras">tensor product</a> of two superalgebras <i>A</i> and <i>B</i> may be regarded as a superalgebra <i>A</i> ⊗ <i>B</i> with a multiplication rule determined by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=(-1)^{|b_{1}||a_{2}|}(a_{1}a_{2}\otimes b_{1}b_{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=(-1)^{|b_{1}||a_{2}|}(a_{1}a_{2}\otimes b_{1}b_{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72795e75806e253340d6c82660682c08e11611f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.118ex; height:3.343ex;" alt="{\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=(-1)^{|b_{1}||a_{2}|}(a_{1}a_{2}\otimes b_{1}b_{2}).}"></span></dd></dl> <p>If either <i>A</i> or <i>B</i> is purely even, this is equivalent to the ordinary ungraded tensor product (except that the result is graded). However, in general, the super tensor product is distinct from the tensor product of <i>A</i> and <i>B</i> regarded as ordinary, ungraded algebras. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations_and_categorical_definition">Generalizations and categorical definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=9" title="Edit section: Generalizations and categorical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can easily generalize the definition of superalgebras to include superalgebras over a commutative superring. The definition given above is then a specialization to the case where the base ring is purely even. </p><p>Let <i>R</i> be a commutative superring. A <b>superalgebra</b> over <i>R</i> is a <a href="/wiki/Supermodule" title="Supermodule"><i>R</i>-supermodule</a> <i>A</i> with a <i>R</i>-bilinear multiplication <i>A</i> × <i>A</i> → <i>A</i> that respects the grading. Bilinearity here means that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\cdot (xy)=(r\cdot x)y=(-1)^{|r||x|}x(r\cdot y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>⋅</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mi>x</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\cdot (xy)=(r\cdot x)y=(-1)^{|r||x|}x(r\cdot y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec1bcd89581ddf08a7a2e23331cedf6f3202347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.787ex; height:3.343ex;" alt="{\displaystyle r\cdot (xy)=(r\cdot x)y=(-1)^{|r||x|}x(r\cdot y)}"></span></dd></dl> <p>for all homogeneous elements <i>r</i> ∈ <i>R</i> and <i>x</i>, <i>y</i> ∈ <i>A</i>. </p><p>Equivalently, one may define a superalgebra over <i>R</i> as a superring <i>A</i> together with an superring homomorphism <i>R</i> → <i>A</i> whose image lies in the supercenter of <i>A</i>. </p><p>One may also define superalgebras <a href="/wiki/Category_theory" title="Category theory">categorically</a>. The <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of all <i>R</i>-supermodules forms a <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal category</a> under the super tensor product with <i>R</i> serving as the unit object. An associative, unital superalgebra over <i>R</i> can then be defined as a <a href="/wiki/Monoid_(category_theory)" title="Monoid (category theory)">monoid</a> in the category of <i>R</i>-supermodules. That is, a superalgebra is an <i>R</i>-supermodule <i>A</i> with two (even) morphisms </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mu &:A\otimes A\to A\\\eta &:R\to A\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>μ</mi> </mtd> <mtd> <mi></mi> <mo>:</mo> <mi>A</mi> <mo>⊗</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mtd> </mtr> <mtr> <mtd> <mi>η</mi> </mtd> <mtd> <mi></mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mu &:A\otimes A\to A\\\eta &:R\to A\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f08864a61a3153fe2052d461253a7d06ce0b0c6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.774ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mu &:A\otimes A\to A\\\eta &:R\to A\end{aligned}}}"></span></dd></dl> <p>for which the usual diagrams commute. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=10" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFKacMartinezZelmanov2001">Kac, Martinez & Zelmanov 2001</a>, p. 3</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFVaradarajan2004">Varadarajan 2004</a>, p. 87</span> </li> <li id="cite_note-deligne-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-deligne_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-deligne_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">See <a rel="nofollow" class="external text" href="http://www.math.ias.edu/QFT/fall/bern-appen1.ps">Deligne's discussion</a> of these two cases.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Superalgebra&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDeligneMorgan1999" class="citation conference cs1"><a href="/wiki/Pierre_Deligne" title="Pierre Deligne">Deligne, P.</a>; Morgan, J. W. (1999). "Notes on Supersymmetry (following Joseph Bernstein)". <i>Quantum Fields and Strings: A Course for Mathematicians</i>. Vol. 1. American Mathematical Society. pp. 41–97. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-2012-5" title="Special:BookSources/0-8218-2012-5"><bdi>0-8218-2012-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Notes+on+Supersymmetry+%28following+Joseph+Bernstein%29&rft.btitle=Quantum+Fields+and+Strings%3A+A+Course+for+Mathematicians&rft.pages=41-97&rft.pub=American+Mathematical+Society&rft.date=1999&rft.isbn=0-8218-2012-5&rft.aulast=Deligne&rft.aufirst=P.&rft.au=Morgan%2C+J.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperalgebra" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKacMartinezZelmanov2001" class="citation book cs1"><a href="/wiki/Victor_Kac" title="Victor Kac">Kac, V. G.</a>; Martinez, C.; <a href="/wiki/Efim_Zelmanov" title="Efim Zelmanov">Zelmanov, E.</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aJHUCQAAQBAJ&q=bibliogroup:%22Graded+simple+Jordan+superalgebras+of+growth+one%22"><i>Graded simple Jordan superalgebras of growth one</i></a>. Memoirs of the AMS Series. Vol. 711. AMS Bookstore. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2645-4" title="Special:BookSources/978-0-8218-2645-4"><bdi>978-0-8218-2645-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graded+simple+Jordan+superalgebras+of+growth+one&rft.series=Memoirs+of+the+AMS+Series&rft.pub=AMS+Bookstore&rft.date=2001&rft.isbn=978-0-8218-2645-4&rft.aulast=Kac&rft.aufirst=V.+G.&rft.au=Martinez%2C+C.&rft.au=Zelmanov%2C+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaJHUCQAAQBAJ%26q%3Dbibliogroup%3A%2522Graded%2Bsimple%2BJordan%2Bsuperalgebras%2Bof%2Bgrowth%2Bone%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperalgebra" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFManin1997" class="citation book cs1"><a href="/wiki/Yuri_Manin" title="Yuri Manin">Manin, Y. I.</a> (1997). <i>Gauge Field Theory and Complex Geometry</i> ((2nd ed.) ed.). Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-61378-1" title="Special:BookSources/3-540-61378-1"><bdi>3-540-61378-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Field+Theory+and+Complex+Geometry&rft.place=Berlin&rft.edition=%282nd+ed.%29&rft.pub=Springer&rft.date=1997&rft.isbn=3-540-61378-1&rft.aulast=Manin&rft.aufirst=Y.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperalgebra" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVaradarajan2004" class="citation book cs1"><a href="/wiki/V._S._Varadarajan" class="mw-redirect" title="V. S. Varadarajan">Varadarajan, V. S.</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sZ1-G4hQgIIC&q=supersymmetry+for+mathematicians&pg=PA1"><i>Supersymmetry for Mathematicians: An Introduction</i></a>. Courant Lecture Notes in Mathematics. Vol. 11. American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3574-6" title="Special:BookSources/978-0-8218-3574-6"><bdi>978-0-8218-3574-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Supersymmetry+for+Mathematicians%3A+An+Introduction&rft.series=Courant+Lecture+Notes+in+Mathematics&rft.pub=American+Mathematical+Society&rft.date=2004&rft.isbn=978-0-8218-3574-6&rft.aulast=Varadarajan&rft.aufirst=V.+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsZ1-G4hQgIIC%26q%3Dsupersymmetry%2Bfor%2Bmathematicians%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASuperalgebra" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output 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href="/wiki/Logic_in_computer_science" title="Logic in computer science">Computational logic</a></li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Computational_statistics" title="Computational statistics">Statistics</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Mathematical_software" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_arbitrary-precision_arithmetic_software" title="List of arbitrary-precision arithmetic software">Arbitrary-precision arithmetic</a></li> <li><a href="/wiki/List_of_finite_element_software_packages" title="List of finite element software packages">Finite element analysis</a></li> <li><a href="/wiki/Tensor_software" title="Tensor software">Tensor software</a></li> <li><a href="/wiki/List_of_interactive_geometry_software" title="List of interactive geometry software">Interactive geometry software</a></li> <li><a href="/wiki/List_of_optimization_software" title="List of optimization software">Optimization software</a></li> <li><a href="/wiki/List_of_statistical_software" title="List of statistical software">Statistical software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical-analysis software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical libraries</a></li> <li><a href="/wiki/Solver" title="Solver">Solvers</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li> <li><a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></li> <li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a></li> <li><a href="/wiki/Clifford_analysis" title="Clifford analysis">Clifford analysis</a> <ul><li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equations</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equations</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/Geometric_analysis" title="Geometric analysis">Geometric analysis</a></li></ul></li> <li><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical systems</a> <ul><li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li></ul></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> <ul><li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></li></ul></li> <li><a href="/wiki/Harmonic_analysis_(mathematics)" class="mw-redirect" title="Harmonic analysis (mathematics)">Harmonic analysis</a> <ul><li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li></ul></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a> <ul><li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Vector_calculus#Vector_algebra" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a> <ul><li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> <ul><li><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical linear algebra</a></li> <li><a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li> <li><a href="/wiki/Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Probability_distribution" title="Probability distribution">Distributions</a> (<a href="/wiki/Random_variable" title="Random variable">random variables</a>)</li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a> / <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">analysis</a></li> <li><a href="/wiki/Functional_integration" title="Functional integration">Path integral</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Stochastic variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical<br />physics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a> <ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a></li> <li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a></li></ul></li> <li><a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">Field theory</a> <ul><li><a href="/wiki/Classical_field_theory" title="Classical field theory">Classical</a></li> <li><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum</a></li> <li><a href="/wiki/Statistical_field_theory" title="Statistical field theory">Statistical</a></li> <li><a href="/wiki/Topological_field_theory" class="mw-redirect" title="Topological field theory">Topological</a></li></ul></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a> <ul><li><a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">in quantum mechanics</a></li></ul></li> <li><a href="/wiki/Potential_theory" title="Potential theory">Potential theory</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a> <ul><li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic</a></li> <li><a href="/wiki/Topological_string_theory" title="Topological string theory">Topological</a></li></ul></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a> <ul><li><a href="/wiki/Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li> <li><a href="/wiki/Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">Supersymmetric theory of stochastic dynamics</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Algebraic_structures" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Algebraic_structures" class="mw-redirect" title="Algebraic structures">Algebraic structures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Feynman integral</a></li> <li><a href="/wiki/Poisson_algebra" title="Poisson algebra">Poisson algebra</a></li> <li><a href="/wiki/Quantum_group" title="Quantum group">Quantum group</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group</a></li> <li><a href="/wiki/Particle_physics_and_representation_theory" title="Particle physics and representation theory">Representation theory</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a class="mw-selflink selflink">Superalgebra</a></li> <li><a href="/wiki/Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Decision_theory" title="Decision theory">Decision sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice theory</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Sociology</a></li> <li>"<a href="/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a>"</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a> <ul><li><a href="/wiki/Japan_Society_for_Industrial_and_Applied_Mathematics" title="Japan Society for Industrial and Applied Mathematics">Japan Society for Industrial and Applied Mathematics</a></li></ul></li> <li><a href="/wiki/Soci%C3%A9t%C3%A9_de_Math%C3%A9matiques_Appliqu%C3%A9es_et_Industrielles" title="Société de Mathématiques Appliquées et Industrielles">Société de Mathématiques Appliquées et Industrielles</a></li> <li><a href="/wiki/International_Council_for_Industrial_and_Applied_Mathematics" title="International Council for Industrial and Applied Mathematics">International Council for Industrial and Applied Mathematics</a></li> <li><a href="/w/index.php?title=European_Community_on_Computational_Methods_in_Applied_Sciences&action=edit&redlink=1" class="new" title="European Community on Computational Methods in Applied Sciences (page does not exist)">European Community on Computational Methods in Applied Sciences</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><a href="/wiki/Category:Mathematics" title="Category:Mathematics">Category</a></b></li> <li><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a> / <a href="/wiki/Topic_outline_of_mathematics" class="mw-redirect" title="Topic outline of mathematics">outline</a> / <a href="/wiki/List_of_mathematics_topics" class="mw-redirect" title="List of mathematics topics">topics list</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Supersymmetry" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Supersymmetry_topics" title="Template:Supersymmetry topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Supersymmetry_topics" title="Template talk:Supersymmetry topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Supersymmetry_topics" title="Special:EditPage/Template:Supersymmetry topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Supersymmetry" style="font-size:114%;margin:0 4em"><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">General topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></li> <li><a href="/wiki/Supersymmetric_gauge_theory" title="Supersymmetric gauge theory">Supersymmetric gauge theory</a></li> <li><a href="/wiki/Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a></li> <li><a href="/wiki/Super_vector_space" title="Super vector space">Super vector space</a></li> <li><a href="/wiki/Supergeometry" title="Supergeometry">Supergeometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Supermathematics" title="Supermathematics">Supermathematics</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Superalgebra</a></li> <li><a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a></li> <li><a href="/wiki/Super-Poincar%C3%A9_algebra" title="Super-Poincaré algebra">Super-Poincaré algebra</a></li> <li><a href="/wiki/Superconformal_algebra" title="Superconformal algebra">Superconformal algebra</a></li> <li><a href="/wiki/Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</a></li> <li><a href="/wiki/Supergroup_(physics)" title="Supergroup (physics)">Supergroup</a></li> <li><a href="/wiki/Superspace" title="Superspace">Superspace</a></li> <li><a href="/wiki/Harmonic_superspace" title="Harmonic superspace">Harmonic superspace</a></li> <li><a href="/wiki/Super_Minkowski_space" title="Super Minkowski space">Super Minkowski space</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supercharge" title="Supercharge">Supercharge</a></li> <li><a href="/wiki/R-symmetry" title="R-symmetry">R-symmetry</a></li> <li><a href="/wiki/Supermultiplet" title="Supermultiplet">Supermultiplet</a></li> <li><a href="/wiki/Short_supermultiplet" title="Short supermultiplet">Short supermultiplet</a></li> <li><a href="/wiki/Bogomol%27nyi%E2%80%93Prasad%E2%80%93Sommerfield_state" title="Bogomol'nyi–Prasad–Sommerfield state">BPS state</a></li> <li><a href="/wiki/Superpotential" title="Superpotential">Superpotential</a></li> <li><a href="/wiki/D-term" title="D-term">D-term</a></li> <li><a href="/wiki/Fayet%E2%80%93Iliopoulos_D-term" title="Fayet–Iliopoulos D-term"> FI D-term</a></li> <li><a href="/wiki/F-term" title="F-term">F-term</a></li> <li><a href="/wiki/Moduli_(physics)" title="Moduli (physics)">Moduli space</a></li> <li><a href="/wiki/Supersymmetry_breaking" title="Supersymmetry breaking">Supersymmetry breaking</a></li> <li><a href="/wiki/Konishi_anomaly" title="Konishi anomaly">Konishi anomaly</a></li> <li><a href="/wiki/Seiberg_duality" title="Seiberg duality">Seiberg duality</a></li> <li><a href="/wiki/Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten theory</a></li> <li><a href="/wiki/Witten_index" title="Witten index">Witten index</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino_gauge" title="Wess–Zumino gauge">Wess–Zumino gauge</a></li> <li><a href="/wiki/Supersymmetric_localization" title="Supersymmetric localization">Localization</a></li> <li><a href="/wiki/Mu_problem" title="Mu problem">Mu problem</a></li> <li><a href="/wiki/Little_hierarchy_problem" title="Little hierarchy problem">Little hierarchy problem</a></li> <li><a href="/wiki/Montonen%E2%80%93Olive_duality" title="Montonen–Olive duality">Electric–magnetic duality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coleman%E2%80%93Mandula_theorem" title="Coleman–Mandula theorem">Coleman–Mandula</a></li> <li><a href="/wiki/Haag%E2%80%93%C5%81opusza%C5%84ski%E2%80%93Sohnius_theorem" title="Haag–Łopuszański–Sohnius theorem">Haag–Łopuszański–Sohnius</a></li> <li><a href="/wiki/Supersymmetry_nonrenormalization_theorems" title="Supersymmetry nonrenormalization theorems">Nonrenormalization</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Field theories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wess%E2%80%93Zumino_model" title="Wess–Zumino model">Wess–Zumino</a></li> <li><a href="/wiki/N_%3D_1_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 1 supersymmetric Yang–Mills theory">N = 1 super Yang–Mills</a></li> <li><a href="/wiki/4D_N_%3D_1_global_supersymmetry" title="4D N = 1 global supersymmetry"> 4D N = 1</a></li> <li><a href="/wiki/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 super Yang–Mills</a></li> <li><a href="/wiki/Super_QCD" title="Super QCD">Super QCD</a></li> <li><a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">MSSM</a></li> <li><a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li> <li><a href="/wiki/6D_(2,0)_superconformal_field_theory" title="6D (2,0) superconformal field theory">6D (2,0) superconformal</a></li> <li><a href="/wiki/ABJM_superconformal_field_theory" title="ABJM superconformal field theory">ABJM superconformal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pure_4D_N_%3D_1_supergravity" title="Pure 4D N = 1 supergravity">Pure 4D N = 1 supergravity</a></li> <li><a href="/wiki/4D_N_%3D_1_supergravity" title="4D N = 1 supergravity">4D N = 1 supergravity</a></li> <li><a href="/wiki/N_%3D_8_supergravity" title="N = 8 supergravity">N = 8 supergravity</a></li> <li><a href="/wiki/Higher_dimensional_supergravity" class="mw-redirect" title="Higher dimensional supergravity">Higher dimensional</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">11D supergravity</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I supergravity</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA supergravity</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB supergravity</a></li> <li><a href="/wiki/Gauged_supergravity" title="Gauged supergravity">Gauged supergravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Superpartner" title="Superpartner">Superpartners</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axino" title="Axino">Axino</a></li> <li><a href="/wiki/Chargino" title="Chargino">Chargino</a></li> <li><a href="/wiki/Gaugino" title="Gaugino">Gaugino</a></li> <li><a href="/wiki/Goldstino" title="Goldstino">Goldstino</a></li> <li><a href="/wiki/Graviphoton" title="Graviphoton">Graviphoton</a></li> <li><a href="/wiki/Graviscalar" title="Graviscalar">Graviscalar</a></li> <li><a href="/wiki/Higgsino" title="Higgsino">Higgsino</a></li> <li><a href="/wiki/Lightest_supersymmetric_particle" title="Lightest supersymmetric particle">LSP</a></li> <li><a href="/wiki/Neutralino" title="Neutralino">Neutralino</a></li> <li><a href="/wiki/R-hadron" title="R-hadron">R-hadron</a></li> <li><a href="/wiki/Sfermion" title="Sfermion">Sfermion</a></li> <li><a href="/wiki/Sgoldstino" title="Sgoldstino">Sgoldstino</a></li> <li><a href="/wiki/Stop_squark" title="Stop squark">Stop squark</a></li> <li><a href="/wiki/Superghost" title="Superghost">Superghost</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Researchers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ian_Affleck" title="Ian Affleck">Affleck</a></li> <li><a href="/wiki/Jonathan_Bagger" title="Jonathan Bagger">Bagger</a></li> <li><a href="/wiki/Marjorie_Batchelor" title="Marjorie Batchelor">Batchelor</a></li> <li><a href="/wiki/Felix_Berezin" title="Felix Berezin">Berezin</a></li> <li><a href="/wiki/Michael_Dine" title="Michael Dine">Dine</a></li> <li><a href="/wiki/Pierre_Fayet" title="Pierre Fayet">Fayet</a></li> <li><a href="/wiki/Jim_Gates" class="mw-redirect" title="Jim Gates">Gates</a></li> <li><a href="/wiki/Yuri_Golfand" title="Yuri Golfand">Golfand</a></li> <li><a href="/wiki/John_Iliopoulos" title="John Iliopoulos">Iliopoulos</a></li> <li><a href="/wiki/Claus_Montonen" title="Claus Montonen">Montonen</a></li> <li><a href="/wiki/David_Olive" title="David Olive">Olive</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Warren_Siegel" title="Warren Siegel">Siegel</a></li> <li><a href="/wiki/Martin_Ro%C4%8Dek" title="Martin Roček">Roček</a></li> <li><a href="/wiki/Alice_Rogers" title="Alice Rogers">Rogers</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Superalgebra&oldid=1238839993">https://en.wikipedia.org/w/index.php?title=Superalgebra&oldid=1238839993</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" 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