Higher trace and Berezinian of matrices over a Clifford algebra

Tiffany Covolo (1) , Valentin Ovsienko (2) , Norbert Poncin (3)

1 Unité de Recherche en Mathématiques
2 ICJ - Institut Camille Jordan
3 Laboratoire de Mathématiques

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 Dieudonné 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.

Domaines: Physique mathématique [math-ph]; Algèbre commutative [math.AC]

Tiffany Covolo, Valentin Ovsienko, Norbert Poncin. Higher trace and Berezinian of matrices over a Clifford algebra. Journal of Geometry and Physics, 2012, 62 (11), pp.2294--2319. ⟨hal-00864657⟩