Higher Trace and Berezinian of Matrices over a Clifford Algebra

TL;DR

This paper introduces new definitions of trace, determinant, and Berezinian for matrices over graded commutative algebras, with applications to quaternionic matrices and a graded version of Liouville's formula.

Contribution

It develops a framework for matrices over (Z_2)^n graded algebras, including a new quaternionic determinant and formulas for Berezinian and trace.

Findings

Graded determinant of even matrices is polynomial in entries.

Recovers classical Dieudonné determinant for quaternions.

Provides explicit Berezinian formula for quaternionic matrices.

Abstract

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonn\'e determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (Z_2)^n graded matrices of degree 0 is polynomial in its entries. In the case of the algebra of quaternions, we calculate the formula for the Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and determinant) by a (Z_2)^n graded version of Liouville's formula.