In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. [1]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.
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Let K be a fixed commutative ring. In most applications, K is a field 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.
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
for all homogeneous elements x and y of A.
A superalgebra is an algebra with a grading (“even” and “odd” elements) such that (i) the bracket of two generators is always antisymmetric except for two odd elements where it is symmetric and (ii) the Jacobi identities are satisfied.[2]
The first of these three identities says that the 0 form a representation of the ordinary Lie algebra spanned by E (Consider the 0 as vectors on which the E act.) The second is equivalent to the first if the Killing form is nonsingular. The last identity restricts the possible representations 0 of the ordinary Lie algebra. This relation is the reason that not every ordinary Lie algebra can be extended to a superalgebra.
Let A be a superalgebra over a commutative ring K. The submodule A0, 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 A1 is an A0-bimodule whose scalar multiplication is just multiplication in A. The product in A equips A1 with a bilinear form
such that
for all x, y, and z in A1. This follows from the associativity of the product in A.
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 xi 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:
The supercommutator on A is the binary operator given by
on homogeneous elements. This can be 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:
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.
The graded tensor product of two superalgebras may be regarded as a superalgebra with a multiplication rule determined by:
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.