and theoretical physics
, a superalgebra
is a Z2
. That is, it is an algebra
over a commutative ring
with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
The prefix super- comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of supermanifolds and superschemes.
Let K be a fixed commutative ring. In most applications, K is a field such as R or C.
A superalgebra over K is an K-module A with a direct sum decomposition
together with a bilinear
where the subscripts are read modulo
A superring, or Z2-graded ring, is a superalgebra over the ring of integers Z.
The elements of Ai 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
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 is one which satisfies a graded version of commutativity. Specifically, A is commutative if