Superalgebra

Let K be a commutative ring. In most applications, K is a field of characteristic 0, such as R or C.

A superalgebra over K is a K-module A with a direct sum decomposition

${\displaystyle A=A_{0}\oplus A_{1}}$

together with a bilinear multiplication A × A → A such that

${\displaystyle A_{i}A_{j}\subseteq A_{i+j}}$

where the subscripts are read modulo 2, i.e. they are thought of as elements of Z₂.

A superring, or Z₂-graded ring, is a superalgebra over the ring of integers Z.

The elements of each of the A_i are said to be homogeneous. The parity of a homogeneous element x, denoted by |x|, is 0 or 1 according to whether it is in A₀ or A₁. Elements of parity 0 are said to be even and those of parity 1 to be odd. If x and y are both homogeneous then so is the product xy and ${\displaystyle |xy|=|x|+|y|}$.

An associative superalgebra is one whose multiplication is associative and a unital superalgebra is one with a multiplicative identity element. 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.

A commutative superalgebra (or supercommutative algebra) is one which satisfies a graded version of commutativity. Specifically, A is commutative if

${\displaystyle yx=(-1)^{|x||y|}xy\,}$

for all homogeneous elements x and y of A. There are superalgebras that are commutative in the ordinary sense, but not in the superalgebra sense. For this reason, commutative superalgebras are often called supercommutative in order to avoid confusion.[2]

• Any algebra over a commutative ring K may be regarded as a purely even superalgebra over K; that is, by taking A₁ to be trivial.
• Any Z- or N-graded algebra may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as tensor algebras and polynomial rings over K.
• In particular, any exterior algebra over K is a superalgebra. The exterior algebra is the standard example of a supercommutative algebra.
• The symmetric polynomials and alternating polynomials together form a superalgebra, being the even and odd parts, respectively. Note that this is a different grading from the grading by degree.
• Clifford algebras are superalgebras. They are generally noncommutative.
• The set of all endomorphisms (denoted ${\displaystyle \mathbf {End} (V)\equiv \mathbf {Hom} (V,V)}$, where the boldface ${\displaystyle \mathrm {Hom} }$ is referred to as internal ${\displaystyle \mathrm {Hom} }$, composed of all linear maps) of a super vector space forms a superalgebra under composition.
• The set of all square supermatrices with entries in K forms a superalgebra denoted by M_p|q(K). This algebra may be identified with the algebra of endomorphisms of a free supermodule over K of rank p|q and is the internal Hom of above for this space.
• Lie superalgebras are a graded analog of Lie algebras. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a universal enveloping algebra of a Lie superalgebra which is a unital, associative superalgebra.

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.

Let R be a commutative superring. A superalgebra over R is a R-supermodule A with a R-bilinear multiplication A × A → A that respects the grading. Bilinearity here means that

${\displaystyle r\cdot (xy)=(r\cdot x)y=(-1)^{|r||x|}x(r\cdot y)}$

for all homogeneous elements r ∈ R and x, y ∈ A.

Equivalently, one may define a superalgebra over R as a superring A together with an superring homomorphism R → A whose image lies in the supercenter of A.

One may also define superalgebras categorically. The category of all R-supermodules forms a monoidal category under the super tensor product with R serving as the unit object. An associative, unital superalgebra over R can then be defined as a monoid in the category of R-supermodules. That is, a superalgebra is an R-supermodule A with two (even) morphisms

${\displaystyle {\begin{aligned}\mu &:A\otimes A\to A\\\eta &:R\to A\end{aligned}}}$

for which the usual diagrams commute.

• Deligne, P.; Morgan, J. W. (1999). "Notes on Supersymmetry (following Joseph Bernstein)". Quantum Fields and Strings: A Course for Mathematicians. Vol. 1. American Mathematical Society. pp. 41–97. ISBN 0-8218-2012-5.

• Kac, V. G.; Martinez, C.; Zelmanov, E. (2001). Graded simple Jordan superalgebras of growth one. Memoirs of the AMS Series. Vol. 711. AMS Bookstore. ISBN 978-0-8218-2645-4.

• Manin, Y. I. (1997). Gauge Field Theory and Complex Geometry ((2nd ed.) ed.). Berlin: Springer. ISBN 3-540-61378-1.

• Varadarajan, V. S. (2004). Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes in Mathematics. Vol. 11. American Mathematical Society. ISBN 978-0-8218-3574-6.