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 -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 Dieudonné determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even -graded matrices of degree 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 -graded version of Liouville’s formula. [Copyright &y& Elsevier]