Superalgebra

In and, a superalgebra is a Z2-. That is, it is an over a  or  with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.

The prefix super- comes from the theory of in theoretical physics. Superalgebras and their representations, s, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called. Superalgebras also play an important role in related field of where they enter into the definitions of s, s and s.

Formal definition
Let K be a. In most applications, K is a of  0, such as R or C.

A superalgebra over K is a A with a  decomposition
 * $$A = A_0\oplus A_1$$

together with a multiplication A &times; A &rarr; A such that
 * $$A_iA_j \sube A_{i+j}$$

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

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

The elements of each of the 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 $$|xy| = |x| + |y|$$.

An associative superalgebra is one whose multiplication is and a unital superalgebra is one with a multiplicative. 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  (or supercommutative algebra) is one which satisfies a graded version of. Specifically, A is commutative if


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

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.

Examples

 * 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- may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as s and s over K.
 * In particular, any over K is a superalgebra. The exterior algebra is the standard example of a.
 * The and  together form a superalgebra, being the even and odd parts, respectively. Note that this is a different grading from the grading by degree.
 * s are superalgebras. They are generally noncommutative.
 * The set of all s (denoted $$\mathbf{End} (V) \equiv \mathbf{Hom}(V,V)$$, where the boldface $$\mathrm {Hom}$$ is referred to as internal $$\mathrm {Hom}$$, composed of all linear maps) of a forms a superalgebra under composition.
 * The set of all square with entries in K forms a superalgebra denoted by Mp(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.
 * s are a graded analog of s. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a of a Lie superalgebra which is a unital, associative superalgebra.

Even subalgebra
Let A be a superalgebra over a commutative ring K. The A0, consisting of all even elements, is closed under multiplication and contains the identity of A and therefore forms a  of A, naturally called the even subalgebra. It forms an ordinary over K.

The set of all odd elements A1 is an A0- whose scalar multiplication is just multiplication in A. The product in A equips A1 with a
 * $$\mu:A_1\otimes_{A_0}A_1 \to A_0$$

such that
 * $$\mu(x\otimes y)\cdot z = x\cdot\mu(y\otimes z)$$

for all x, y, and z in A1. This follows from the associativity of the product in A.

Grade involution
There is a canonical  on any superalgebra called the grade involution. It is given on homogeneous elements by
 * $$\hat x = (-1)^{|x|}x$$

and on arbitrary elements by
 * $$\hat x = x_0 - x_1$$

where xi are the homogeneous parts of x. If A has no (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of A:
 * $$A_i = \{x \in A : \hat x = (-1)^i x\}.$$

Supercommutativity
The  on A is the binary operator given by
 * $$[x,y] = xy - (-1)^{|x||y|}yx$$

on homogeneous elements, extended to all of A by linearity. Elements x and y of A are said to supercommute if [x, y] = 0.

The supercenter of A is the set of all elements of A which supercommute with all elements of A:
 * $$\mathrm{Z}(A) = \{a\in A : [a,x]=0 \text{ for all } x\in A\}.$$

The supercenter of A is, in general, different than the of A as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of A.

Super tensor product
The graded of two superalgebras A and B may be regarded as a superalgebra A &otimes; B with a multiplication rule determined by:
 * $$(a_1\otimes b_1)(a_2\otimes b_2) = (-1)^{|b_1||a_2|}(a_1a_2\otimes b_1b_2).$$

If either A or B 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 A and B regarded as ordinary, ungraded algebras.

Generalizations and categorical definition
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 A with a R-bilinear multiplication A &times; A &rarr; A that respects the grading. Bilinearity here means that
 * $$r\cdot(xy) = (r\cdot x)y = (-1)^{|r||x|}x(r\cdot y)$$

for all homogeneous elements r &isin; R and x, y &isin; A.

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

One may also define superalgebras. The of all R-supermodules forms a  under the super tensor product with R serving as the unit object. An associative, unital superalgebra over R can then be defined as a in the category of R-supermodules. That is, a superalgebra is an R-supermodule A with two (even) morphisms
 * $$\begin{align}\mu &: A\otimes A \to A\\ \eta &: R\to A\end{align}$$

for which the usual diagrams commute.