Higher Trace and Berezinian of Matrices over a Clifford Algebra
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.
Keywords
Cite
@article{arxiv.1109.5877,
title = {Higher Trace and Berezinian of Matrices over a Clifford Algebra},
author = {Tiffany Covolo and Valentin Ovsienko and Norbert Poncin},
journal= {arXiv preprint arXiv:1109.5877},
year = {2014}
}
Comments
36 pages; added references, corrected typos