supertrace

Supertrace is a generalization of the ordinary trace for Z2-graded vector spaces. Let V = V0 ⊕ V1 be a finite-dimensional vector space over a field of characteristic not two, equipped with a decomposition into even and odd parts. An endomorphism T of V can be written in block form as T = [T00 T01; T10 T11], where T00: V0→V0, T01: V1→V0, T10: V0→V1, and T11: V1→V1. The supertrace of T is defined by str(T) = Tr(T00) − Tr(T11). This extends linearly to all endomorphisms and is often called the graded trace.

Key properties include linearity and a form of cyclicity adapted to the grading. In particular, str([X,Y]) = 0 for all X,Y in the endomorphism algebra End(V), where [X,Y] denotes the supercommutator XY − (−1)^{|X||Y|} YX. The supertrace reduces to the ordinary trace when V is purely even (V1 = 0). It also yields the superdimension sd(V) = str(Id_V) = dim(V0) − dim(V1).

In representation theory, the supertrace plays the role of a character for representations of Lie superalgebras, assigning to a representation the function g ↦ str(ρ(g)). In differential geometry and mathematical physics, the supertrace appears in index theory and in formulations of supersymmetric theories, such as the McKean–Singer formula where the index is expressed as a supertrace of a heat operator.

Variants include the graded trace on more general Z2-graded algebras and their representations. The concept captures the balance between bosonic (even) and fermionic (odd) degrees of freedom in a concise, algebraically coherent way.

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