superalgebra

A superalgebra is a Z2-graded algebra over a field, usually denoted k, equipped with a decomposition A = A0 ⊕ A1 into even and odd parts. The multiplication respects the grading in the sense that Ai Aj ⊆ A{i+j mod 2}. Homogeneous elements have parity |a| ∈ {0,1}, and the product of homogeneous elements is determined by their degrees. In a general associative superalgebra, associativity holds and the grading is preserved by multiplication. A notable specialization is the supercommutative algebra, where for homogeneous a,b we have ab = (−1)^{|a||b|} ba; this generalizes ordinary commutativity by introducing signs whenever odd elements are exchanged.

Key examples include the exterior algebra Λ(V) of a vector space V, which is naturally Z2-graded and supercommutative: its odd generators anticommute and square to zero. The algebra of endomorphisms End(V) of a supervector space V = V0 ⊕ V1 gives the matrix superalgebra M(m|n), with its even part consisting of parity-preserving endomorphisms and its odd part of parity-flipping ones.

Lie superalgebras generalize Lie algebras to the graded setting. A Lie superalgebra g is a Z2-graded vector space equipped with a bilinear bracket [ , ] of degree 0 that is graded antisymmetric and satisfies the graded Jacobi identity. The universal enveloping algebra U(g) encodes representations of g. A supermodule (or graded module) over a superalgebra carries a compatible Z2-grading describing its action.

Applications of superalgebras appear in mathematics and theoretical physics, notably in supersymmetry and supergeometry, where they provide a framework to unify bosonic and fermionic structures.