supercommutativity

Supercommutativity is a property of algebras equipped with a Z/2 grading, known as superalgebras. Such an algebra A decomposes into A = A0 ⊕ A1, where elements of A0 are considered even and elements of A1 odd. For homogeneous elements a and b with degrees |a|, |b| ∈ {0,1}, A is called supercommutative if the multiplication satisfies ab = (-1)^{|a||b|} ba. Equivalently, the graded commutator [a, b] = ab - (-1)^{|a||b|} ba vanishes for all homogeneous elements. The relation extends by linearity to all elements.

Examples and implications: The exterior algebra Λ(V) is a fundamental example of a supercommutative algebra, because the product of two odd elements anticommutes: x ∧ y = - y ∧ x. A purely even algebra, concentrated in degree 0, is automatically supercommutative and reduces to ordinary commutativity. In many constructions, the even part commutes with all elements, while the odd part anticommutes with other odd elements. The concept generalizes ordinary commutativity by incorporating a sign determined by parity.

Applications and context: Supercommutativity plays a central role in the theory of superalgebras, supergeometry, and related areas of mathematical physics. In physics, it underpins the statistics of fields in supersymmetric theories: bosonic (even) quantities commute with others, while fermionic (odd) quantities acquire signs upon exchange. The graded commutator and related notions of superderivations and tensor products are standard tools in this framework. See also graded algebra, superalgebra, and graded geometry.