Supertrace

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,

then the supertrace

Let us show that the supertrace does not depend on a basis. Suppose e₁, ..., e_p are the even basis vectors and e_p+1, ..., e_p+q are the odd basis vectors. Then, the components of T, which are elements of A, are defined as

The grading of Tⁱ_j is the sum of the gradings of T, e_i, e_j mod 2.

A change of basis to e_1', ..., e_p', e_(p+1)', ..., e_(p+q)' is given by the supermatrix

and the inverse supermatrix

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

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

In the case where T is odd, we have

The ordinary trace is not basis independent, so the appropriate trace to use in the Z₂-graded setting is the supertrace.

The supertrace satisfies the property

for all T₁, T₂ in End(V). In particular, the supertrace of a supercommutator is zero.

In fact, one can define a supertrace more generally for any associative superalgebra E over a commutative superalgebra A as a linear map tr: E -> A which vanishes on supercommutators. Such a supertrace is not uniquely defined; it can always at least be modified by multiplication by an element of A.

In supersymmetric quantum field theories, in which the action integral is invariant under a set of symmetry transformations (known as supersymmetry transformations) whose algebras are superalgebras, the supertrace has a variety of applications. In such a context, the supertrace of the mass matrix for the theory can be written as a sum over spins of the traces of the mass matrices for particles of different spin:

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