In mathematics and theoretical physics, a superalgebra is a <math>\mathbb{Z}_2</math>-graded algebra. That is, it is an algebra over a commutative ring or field 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 graded manifolds, supermanifolds and superschemes.
Formal definition
Let <math>K</math> be a commutative ring. In most applications, <math>K</math> is a field of characteristic 0, such as <math>\mathbb{R}</math> or <math>\mathbb{C}</math>.
A superalgebra over <math>K</math> is a <math>K</math>-module <math>A</math> with a direct sum decomposition
:<math>A = A_0\oplus A_1</math>
together with a bilinear multiplication <math>A\times A\to A</math> such that
:<math>A_iA_j \sube A_{i+j}</math>
where the subscripts are read modulo 2, i.e. they are thought of as elements of <math>\mathbb{Z}_2</math>.
A superring, or <math>\mathbb{Z}_2</math>-graded ring, is a superalgebra over the ring of integers <math>\mathbb{Z}</math>.
The elements of each of the <math>A_i</math> are said to be homogeneous. The parity of a homogeneous element <math>x</math>, denoted by <math>|x|</math>, is 0 or 1 according to whether it is in <math>A_0</math> or <math>A_1</math>. Elements of parity 0 are said to be even and those of parity 1 to be odd. If <math>x</math> and <math>y</math> are both homogeneous, then so is the product <math>xy</math> and <math>|xy| = |x| + |y|</math>.
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, <math>A</math> is commutative if
:<math>yx = (-1)^{|x||y|}xy\,</math>
for all homogeneous elements <math>x</math> and <math>y</math> of <math>A</math>. 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.
Sign conventions
When the <math>\mathbb{Z}_2</math> grading arises as a "rollup" of a <math>\mathbb{Z}</math>- or <math>\mathbb{N}</math>-graded algebra into even and odd components, then two distinct (but essentially equivalent) sign conventions can be found in the literature. 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
:<math>xy\mapsto (-1)^{mn+pq} yx</math>
where <math>m=\deg x</math> is the degree (<math>\mathbb{Z}</math>- or <math>\mathbb{N}</math>-grading) of <math>x</math> and <math>p</math> the parity. Likewise, <math>n=\deg y</math> is the degree of <math>y</math> and with parity <math>q.</math> This convention is commonly seen in conventional mathematical settings, such as differential geometry and differential topology. The other convention is to take
:<math>xy\mapsto (-1)^{pq} yx</math>
with the parities given as <math>p=m\bmod 2</math> and <math>q=n\bmod 2</math> 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 Pierre Deligne.