In mathematics and theoretical physics, a superalgebra-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

be a commutative ring is a field of characteristic.

A superalgebra is a K-module with a direct sum decomposition

together with a bilinear such that

where the subscripts are read modulo.

A superring-graded ring, is a superalgebra over the ring of integers.

are said to be homogeneous. The parity. Elements of parity 0 are said to be even and those of parity 1 to be odd.

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 is commutative if

. 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

-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

This convention is commonly seen in conventional mathematical settings, such as differential geometry and differential topology. The other convention is to take

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.

Examples

• to be the trivial algebra (the algebra with one element).

• -graded algebra may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as tensor algebras and polynomial rings.

• In particular, any exterior algebra is a superalgebra. The exterior algebra is the standard example of a supercommutative algebra.

• The symmetric polynomials and alternating polynomials-graded algebra of polynomials, where the grading is by degrees.

• Clifford algebras are superalgebras. Clifford algebras for low-dimensional orthogonal spaces such as projective geometric algebra offer some visual intuition for superalgebras: odd elements correspond to handedness-reversing isometries of the space such as rotoreflections; even elements correspond to handedness-preserving ones such as rotations and screw motions.

• The set of all endomorphisms is referred to as internal, composed of all linear maps) of a super vector space forms a superalgebra under composition.

• The set of all square supermatrices and is the internal Hom of above for this space.

• Lie superalgebras are a graded analog of Lie algebras. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a universal enveloping algebra of a Lie superalgebra which is a unital, associative superalgebra.

Further definitions and constructions

Even subalgebra

. The submodule and therefore forms a subalgebra, naturally called the even subalgebra. It forms an ordinary algebra.

-bimodule with a bilinear form

such that

.

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

has no 2-torsion:

Supercommutativity

The supercommutator is the binary operator given by

are said to supercommute.

The supercenter:

is, in general, different than the center.

Super tensor product

The graded tensor product with a multiplication rule determined by:

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.

be a commutative superring. A superalgebra is an R-supermodule that respects the grading. Bilinearity here means that

.

.

One may also define superalgebras categorically. The category-supermodules forms a monoidal category can then be defined as a monoid with two (even) morphisms

for which the usual diagrams commute.

Notes

References