Superalgebra

In mathematics and theoretical physics, a superalgebra is a Z2-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 "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
together with a bilinear multiplication "A" × "A" → "A" such that
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 , is 0 or 1 according to whether it is in "A"0 or "A"1. 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 formula_3.
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
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.
Further definitions and constructions.
Even subalgebra.
Let "A" be a superalgebra over a commutative ring "K". The submodule "A"0, consisting of all even elements, is closed under multiplication and contains the identity of "A" and therefore forms a subalgebra of "A", naturally called the even subalgebra. It forms an ordinary algebra over "K".
The set of all odd elements "A"1 is an "A"0-bimodule whose scalar multiplication is just multiplication in "A". The product in "A" equips "A"1 with a bilinear form
such that
for all "x", "y", and "z" in "A"1. This follows from the associativity of the product in "A".
Grade involution.
There is a canonical involutive automorphism on any superalgebra called the grade involution. It is given on homogeneous elements by
and on arbitrary elements by
where "x""i" are the homogeneous parts of "x". If "A" has no 2-torsion (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of "A":
Supercommutativity.
The supercommutator on "A" is the binary operator given by
on homogeneous elements, extended to all of "A" by linearity. Elements "x" and "y" of "A" are said to supercommute if .
The supercenter of "A" is the set of all elements of "A" which supercommute with all elements of "A":
The supercenter of "A" is, in general, different than the center of "A" as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of "A".
Super tensor product.
The graded tensor product of two superalgebras "A" and "B" may be regarded as a superalgebra "A" ⊗ "B" with a multiplication rule determined by:
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 "R"-supermodule "A" with a "R"-bilinear multiplication "A" × "A" → "A" that respects the grading. Bilinearity here means that
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
for which the usual diagrams commute.