Z2grading

Z2grading refers to a Z2-grading, a way of decomposing a mathematical object into two pieces indexed by the two elements of the cyclic group Z2 = {0,1}. For a vector space V, a Z2-grading is a direct sum V = V0 ⊕ V1, where V0 contains even elements and V1 contains odd elements. The parity of a homogeneous element v ∈ V_i is i ∈ {0,1}. A linear map that preserves the grading is even; a map that shifts parity is odd.

In the context of algebras, a Z2-graded algebra (or superalgebra) A has a decomposition A = A0 ⊕ A1 with A_i A_j ⊆ A_{i+j mod 2}. The product of homogeneous elements a ∈ A_i and b ∈ A_j has degree i+j mod 2. Graded-commutativity means ab = (-1)^{|a||b|} ba for homogeneous a,b, with |a| the degree.

Common examples include the exterior algebra Λ(V) of a vector space V, which is naturally Z2-graded by even and odd forms, and the endomorphism algebra of a Z2-graded space. Lie superalgebras and many constructions in supergeometry rely on Z2-gradings.

Applications extend to representation theory, homological algebra, and theoretical physics, especially supersymmetry and BRST cohomology, where parity distinguishes bosonic and fermionic sectors.

Notes: Z2-grading is often denoted by parity or by grading over Z/2Z; it is distinct from integer Z-gradings, which assign degrees in all integers rather than mod 2.