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arXiv · 1109.5877

Higher Trace and Berezinian of Matrices over a Clifford Algebra

Tiffany Covolo·Valentin Ovsienko·Norbert Poncin
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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.

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Tiffany Covolo, Valentin Ovsienko, Norbert Poncin. 2011-09-27. Higher Trace and Berezinian of Matrices over a Clifford Algebra. https://doi.org/10.1016/j.geomphys.2012.07.004

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