Homogeneous element

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 supermanifolds and superschemes.

Formal definition

Let K be a fixed commutative ring. In most applications, K is a field such as R or C.

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

A = A_0oplus A_1

together with a bilinear multiplication A × A → A such that

A_iA_j sube A_{i+j}

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 A_i 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 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

yx = (-1)^

xy,>

for all homogeneous elements x and y of A.

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-graded algebra may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as tensor algebras and polynomial rings over K.
In particular, any exterior algebra over K is a superalgebra. The exterior algebra is the standard example of a supercommutative algebra.
Clifford algebras are superalgebras. They are generally noncommutative.
The set of all endomorphisms (both even and odd) of a super vector space forms a superalgebra under composition.
The set of all square supermatrices with entries in K forms a superalgebra denoted by M_p|q(K). This algebra may be identified with the algebra of endomorphisms of a free supermodule over K of rank p|q.
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

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 a A₀-bimodule whose scalar multiplication is just multiplication in A. The product in A equips A₁ with a bilinear form

mu:A_1otimes_{A_0}A_1 to A_0

such that

mu(xotimes y)cdot z = xcdotmu(yotimes z)

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

Grade involution

The is a canonical involutive automorphism on any superalgebra called the grade involution. It is given on homogeneous elements by

hat x = (-1)^

x> and on arbitrary elements by

hat x = x_0 - x_1

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:

A_i = {x in A : hat x = (-1)^i x}.

Supercommutativity

The supercommutator on A is the binary operator given by

[x,y] = xy - (-1)^

yx,> 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:

Z(A) = {ain A : [a,x]=0 text{ for all } xin 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 may be regarded as a superalgebra with a multiplication rule determined by:

(a_1otimes b_1)(a_2otimes b_2) = (-1)^

(a_1a_2otimes b_1b_2).>

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

rcdot(xy) = (rcdot x)y = (-1)^

a_2

>x(rcdot y) 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

begin{align}mu &: Aotimes A to A eta &: Rto Aend{align}

for which the usual diagrams commute.

References

Deligne, Pierre; John W. Morgan (1999). "Notes on Supersymmetry (following Joseph Bernstein)". Quantum Fields and Strings: A Course for Mathematicians, 41–97. ISBN 0-8218-2012-5. .
Manin, Y. I. (1997). Gauge Field Theory and Complex Geometry. (2nd ed.), Berlin: Springer.
Varadarajan, V. S. (2004). Supersymmetry for Mathematicians: An Introduction. American Mathematical Society. ISBN 0-8218-3574-2.