Higher trace and Berezinian of matrices over a Clifford algebra |
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Authors: | Tiffany Covolo Valentin Ovsienko Norbert Poncin |
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Affiliation: | 1. Université du Luxembourg, Unité de Recherche en Mathématiques, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg;2. CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex, France |
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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 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. |
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Keywords: | 17A70 58J52 58A50 15A66 11R52 |
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