View source for Supersymmetry

The first appearance of supersymmetry is Grassmann's definition of an algebra that, although non-commutative, is commutative up to a sign factor. The Grassmann algebra (or exterior algebra) of a vector space is perhaps the earliest example of a class of algebras, supercommutative associative algebras, which appear extensively in topology (e.g. cohomology algebras, Steenrod algebras, and Hopf algebras (in the sense of [[#References|[a13]]]), cf. also [[Steenrod algebra|Steenrod algebra]]; [[Hopf algebra|Hopf algebra]]), and in geometry (e.g., the de Rham complex; cf. also [[De Rham cohomology|de Rham cohomology]]). Lie superalgebras, the analogous generalization of Lie algebras (cf. also [[Superalgebra|Superalgebra]]), first appeared in [[Deformation|deformation]] theory, in a geometric context, in the work of A. Frölicher and A. Nijenhuis in the late 1950s, and, in an algebraic context, in the work of M. Gerstenhaber in the 1960s. In the early 1970s, Lie superalgebras and Lie supergroups arose in physics, where they were used to describe transformations (symmetries) connecting bosons, which are described by commuting variables, and fermions, which are described by anti-commuting variables. For more details, see [[#References|[a5]]], [[#References|[a3]]], [[#References|[a6]]], [[#References|[a16]]] and the original articles cited therein.

A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s1203202.png" />-graded vector space, or super vector space, is a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s1203203.png" /> which is a direct sum of two components, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s1203204.png" />. Vectors belonging to the summands are called homogeneous, and the parity function, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s1203205.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s1203206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s1203207.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s1203208.png" />, distinguishes between those in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s1203209.png" />, called even vectors, and those in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032010.png" />, called odd vectors. In the super vector space representing physical particles, the even vectors represent bosons and the odd vectors represent fermions.

The term [[Superalgebra|superalgebra]] is used for a super vector space with a bilinear product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032011.png" /> that preserves the grading, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032012.png" />. For example, a Lie superalgebra is a super vector space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032013.png" />, together with a bracket preserving the grading, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032014.png" /> is a super derivation. In other words,

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Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032017.png" /> is a [[Lie algebra|Lie algebra]] in the standard sense.

The Lie superalgebra first used by physicists, the physical supersymmetry algebra, has an even component which is the standard Poincaré algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032018.png" /> (cf. also [[Contraction of a Lie algebra|Contraction of a Lie algebra]]). The odd component consists of spinors, which, when represented as symmetries of physical fields, transform fermions to bosons, and conversely, thus defining a  "supersymmetry" . V.G. Kac [[#References|[a9]]] has given a complete classification of the finite-dimensional simple Lie superalgebras over the complex numbers. There are two essentially different types: classical and Cartan superalgebras. The classical simple superalgebras, characterized by having an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032019.png" /> which is reductive (cf. also [[Lie algebra, reductive|Lie algebra, reductive]]), include the Wess–Zumino spin-conformal Lie superalgebra, which extends the physical supersymmetry algebra described above, see [[#References|[a6]]]. The Cartan superalgebras have infinite-dimensional generalizations which play an important role in conformal field theory, see [[#References|[a1]]] and references therein.

In categorical terms, super vector spaces form a symmetric monoidal [[Abelian category]] with [[Monoidal category|monoidal structure]] defined by

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and

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and with symmetry defined by

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The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032023.png" /> appears in any supersymmetric analogue of a classical formula which involves moving one element past another. The basic rule, sometimes called the supersymmetric  "sign convention" , states that a factor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032024.png" /> appears whenever two elements, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032026.png" /> are flipped. (The sign convention does not always lead to a straightforward supersymmetric analogue, see the discussion of superdeterminants below.) The symmetric groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032027.png" /> act on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032028.png" />th tensor power of a super vector space and there exist super analogues of the usual representation theory, e.g. Schur's double centralizer theorem, Young diagrams [[#References|[a4]]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032029.png" /> is an associative superalgebra with product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032030.png" />, the supercommutator is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032031.png" /> for homogeneous elements, and extended linearly. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032032.png" /> is supercommutative if and only if the supercommutator is identically zero. In characteristic other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032033.png" />, this implies that odd elements are nilpotent of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032034.png" />. The symmetric algebra of a super vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032035.png" /> is defined as the quotient of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032036.png" /> by the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032037.png" />. Since the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032038.png" /> inherits a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032039.png" />-grading, so does the quotient by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032040.png" /> graded ideal, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032041.png" />, and one has:

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032043.png" /> is totally even, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032045.png" /> is the usual symmetric algebra. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032046.png" /> is totally odd, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032047.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032048.png" /> is the Grassmann algebra (cf. also [[Cartan method of exterior forms|Cartan method of exterior forms]]), which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032049.png" />-graded and supercommutative. Thus, the supersymmetric world unifies the standard concepts of commutativity and skew commutativity.

Another important example of an associative superalgebra is the [[Universal enveloping algebra|universal enveloping algebra]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032050.png" />, of a Lie superalgebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032051.png" />,

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It is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032053.png" />-graded but not supercommutative. As in the standard case, it has an increasing filtration, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032054.png" />th level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032055.png" /> equal to the subspace generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032056.png" />-fold products of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032057.png" />. There is an analogue of the Poincaré–Birkhoff–Witt theorem (cf. [[Birkhoff–Witt theorem|Birkhoff–Witt theorem]]), stating that the associated graded algebra is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032058.png" />; see [[#References|[a6]]].

A module over a superalgebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032059.png" />, is assumed to have a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032060.png" />-grading which is compatible with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032061.png" />-action. As in the classical case, every left (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032062.png" />-graded) module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032063.png" /> over a supercommutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032064.png" /> becomes naturally an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032065.png" />-bimodule under the right action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032066.png" />. The parity exchange functor is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032068.png" />. The free finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032069.png" />-module of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032070.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032071.png" />. Since the left and right module structures do not coincide, one has to distinguish between left and right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032072.png" />-module morphisms. After settling for a convention, say of considering the (symmetric monoidal) category of right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032073.png" />-modules, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032074.png" />. Then, using right coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032075.png" /> and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032076.png" />, the action of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032077.png" /> on an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032078.png" /> is given by left multiplication of the corresponding matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032079.png" />, with the column vector of right coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032080.png" />. The matrix,

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is even (odd) if the submatrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032083.png" /> have even (odd) coefficients and the submatrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032085.png" /> have odd (even) coefficients, respectively. In defining the dual module, one must distinguish again between left and right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032086.png" />-module morphisms into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032087.png" />. Having fixed one choice of dual, say dual on the left, then, as usual, given a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032088.png" /> there is dual morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032089.png" /> with the property

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In terms of matrices, this leads to the definition a supertranspose, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032091.png" />. However, in contrast to the usual transpose, the supertranspose is not an involution but has order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032092.png" />, see [[#References|[a15]]].

The supertrace is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032093.png" /> (square matrices acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032094.png" />) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032095.png" />, and has the same properties as the usual trace (cf. also [[Trace of a square matrix|Trace of a square matrix]]). It is a morphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032096.png" />-modules, vanishes on supercommutators, and is invariant under the supertranspose. The fundamental difference is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032097.png" />.

A definition of superdeterminant is more complicated. In the standard case the determinant of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032098.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s12032099.png" /> can be defined using the induced action on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320100.png" />th exterior power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320101.png" /> of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320102.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320103.png" /> operates. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320104.png" />, the induced action is multiplication by a scalar, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320105.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320106.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320107.png" />-graded and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320108.png" />, there is no longer a top exterior power of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320109.png" /> and the standard approach will not work. F.A. Berezin has given a definition of a superdeterminant (now called the Berezinian) for invertible even endomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320110.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320111.png" /></td> </tr></table>

It has the important, and not obvious, property of being a multiplicative homomorphism. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320112.png" />. It is possible to define the Berezinian in the spirit of the classical definition using the [[Koszul complex|Koszul complex]], see [[#References|[a15]]]. Other possible definitions have been studied by I. Kantor, [[#References|[a10]]].

Supergeometry begins with the definition of a real (complex) superdomain of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320113.png" />, which is a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320114.png" /> consisting of an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320115.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320116.png" />), a supercommutative superalgebra of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320117.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320118.png" /> is the appropriate algebra of functions (smooth, analytic, etc.), and an evaluation mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320119.png" /> defined for pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320123.png" />. A mapping of superdomains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320124.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320125.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320126.png" />, is a mapping of topological spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320127.png" /> is a mapping of superalgebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320128.png" />. A super-manifold (cf., also [[Super-manifold|Super-manifold]]) is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320129.png" /> consisting of a [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320130.png" /> and a [[Sheaf|sheaf]] of supercommutative algebras, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320131.png" />, which is locally isomorphic to a superdomain. Starting with these basic definitions one can develop a theory of super-manifolds analogous to the classical theory, see [[#References|[a2]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a15]]]. Supersymmetric Yang–Mills theory [[#References|[a15]]], [[#References|[a18]]] and the theory of superstrings [[#References|[a8]]] (which may be the key to a unified theory of the fundamental forces of nature) have stimulated an explosion of new ideas in mathematics, in particular in the study of invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320133.png" />-manifolds, see [[#References|[a2]]], [[#References|[a17]]].

The theory of [[Quantum groups|quantum groups]] has led to an extended concept of symmetry for modules over a [[Quasi-triangular Hopf algebra|quasi-triangular Hopf algebra]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320134.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320135.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320136.png" /> and braiding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320137.png" />. In the triangular case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320138.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320139.png" />, the braiding is a symmetry and there is a theory of quantum commutativity [[#References|[a7]]]. For general Hopf algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320140.png" /> (cf. also [[Hopf algebra|Hopf algebra]]), there is a theory of  "H-commutativity" , related to Yetter–Drinfel'd categories, [[#References|[a14]]], [[#References|[a19]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.G. Kac,   "Classification of infinite-dimensional simple linearly compact Lie superalgebras"  ''Adv. Math.'' , '''139'''  (1998)  pp. 1–55</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Bernstein,   "Lectures on supersymmetry"  P. Deligne (ed.)  P. Etingof (ed.)  D. Freed (ed.)  L. Jeffrey (ed.)  D. Kazhdan (ed.)  D. Morrison (ed.)  E. Witten (ed.) , ''Quantum Fields and Strings: A Course for Mathematicians'' , Amer. Math. Soc.  (to appear)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F.A. Berezin,   G.I. Kats,   "Lie groups with noncommuting parameters"  ''Mat. Sb. USSR'' , '''11'''  (1970)  pp. 311–320</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Berele,   A. Regev,   "Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras"  ''Adv. Math.'' , '''64''' :  2  (1987)  pp. 118–175</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  F.A. Berezin,   "The mathematical basis of supersymmetric field theories"  ''Soviet J. Nucl. Phys.'' , '''29'''  (1979)  pp. 857–866</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Corwin,   Y. Ne'eman,   S. Sternberg,   "Graded Lie algebras in mathematics and physics"  ''Rev. Mod. Phys.'' , '''47'''  (1975)  pp. 573–604</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Cohen,   S. Westreich,   "From supersymmetry to quantum commutativity"  ''J. Algebra'' , '''168'''  (1994)  pp. 1–27</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M.B. Green,   J.H. Schwarz,   E. Witten,   "Superstring Theory" , ''Mon. Math. Phys.'' , Cambridge Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  V.G. Kac,   "Lie superalgebras"  ''Adv. Math.'' , '''26'''  (1977)  pp. 8–96</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Kantor,   "On the concept of determinant in the supercase"  ''Commun. Algebra'' , '''22''' :  10  (1994)  pp. 3679–3739</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  B. Kostant,   "Graded manifolds, graded Lie theory and prequantization" , ''Differential Geom. Methods in Math. Phys. Proc. Symp. Bonn 1975'' , ''Lecture Notes Math.'' , '''570''' , Springer  (1977)  pp. 177–306</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  D.A. Leites,   "Introduction to the theory of supermanifolds"  ''Russian Math. Surveys'' , '''35'''  (1980)  pp. 3–57</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  J. Milnor,   J. Moore,   "On the Structure of Hopf algebras"  ''Ann. of Math.'' , '''81'''  (1965)  pp. 211–264</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  S. Majid,   "Foundations of quantum group theory" , Cambridge Univ. Press  (1995)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  Yu.I. Manin,   "Gauge field theory and complex geometry" , Springer  (1984)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  V.I. Ogievetski,   L. Mezinchesku,   "Boson-fermion symmetries and superfields"  ''Soviet Phys. Uspekhi'' , '''18''' :  12  (1975)  pp. 960–981</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  N. Seiberg,   E. Witten,   "Monopoles, duality, and chiral symmetry breaking in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120320/s120320142.png" /> supersymmetric QCD"  ''Nucl. Phys. B'' , '''431'''  (1994)  pp. 581–640</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  S. Shnider,   R.O. Wells Jr.,   "Supermanifolds, Super Twistor Spaces and Super Yang–Mills Fields" , ''Sém. Math. Sup.'' , Les Presses de l'Univ. Montréal  (1989)</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  D. Yetter,   "Quantum groups and representations of monoidal categories"  ''Math. Proc. Cambridge Philos. Soc.'' , '''108'''  (1990)  pp. 261–290</TD></TR></table>