HIGHER TRACE AND BEREZINIAN

ar X iv :1 10 9. 58 77 v1 [ ma th. DG ] 27 Se p 2 01 1 HIGHER TRACE AND BEREZINIAN OF MATRICES OVER A CLIFFORD ALGEBRA TIFFANY COVOLO VALENTIN OVSIENKO NORBERT PONCIN Abstract. We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z2)n-graded commutative associative algebra A. 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 Dieudonne determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (Z2)n-graded matrices of degree 0 is polynomial in its entries. In the case of the algebra A = H 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 (Z2)n-graded version of Liouville's formula. Contents 1. Introduction 2 2. (Z2)n-Graded Algebra 4 2.1. General Notions 4 2.2. (Z2)n- and (Z2)n+1-Grading on Clifford Algebras 7 3.

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