What is "supertrace"

n. (context mathematics English) A specific form of a trace in a superalgebra

In the theory of superalgebras, if A is a commutative superalgebra, V is a free right A- supermodule and T is an endomorphism from V to itself, then the supertrace of T, str(T) is defined by the following trace diagram:

More concretely, if we write out T in block matrix form after the decomposition into even and odd subspaces as follows,

$$T=\begin{pmatrix}T_{00}&T_{01}\\T_{10}&T_{11}\end{pmatrix}$$

then the supertrace

str(T) = the ordinary trace of T − the ordinary trace of T.

Let us show that the supertrace does not depend on a basis. Suppose e, ..., e are the even basis vectors and e, ..., e are the odd basis vectors. Then, the components of T, which are elements of A, are defined as

T(e) = eT. 

The grading of T is the sum of the gradings of T, e, e mod 2.

A change of basis to e, ..., e, e, ..., e is given by the supermatrix

e = eA

and the inverse supermatrix

e = e(A),  

where of course, AA = AA = 1 (the identity).

We can now check explicitly that the supertrace is basis independent. In the case where T is even, we have

:\operatorname{str}(A^{-1} T A)=(-1)^{|i'|} (A^{-1})^{i'}_j T^j_k A^k_{i'}=(-1)^{|i'|}(-1)^{(|i'|+|j|)(|i'|+|j|)}T^j_k A^k_{i'} (A^{-1})^{i'}_j=(-1)^{|j|} T^j_j