Higher trace and Berezinian of matrices over a Clifford algebra

Covolo, Tiffany; Ovsienko, Valentin; Poncin, Norbert

doi:10.1016/j.geomphys.2012.07.004

Reference : Higher trace and Berezinian of matrices over a Clifford algebra


	Document type :	Scientific journals : Article
	Discipline(s) :	Physical, chemical, mathematical & earth Sciences : Mathematics
	To cite this reference:	http://hdl.handle.net/10993/9369

Title :	Higher trace and Berezinian of matrices over a Clifford algebra
Language :	English
Author, co-author :	Covolo, Tiffany [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
	Ovsienko, Valentin [CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1]
	Poncin, Norbert [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Publication date :	2012
Journal title :	Journal of Geometry and Physics
Publisher :	Elsevier
Volume :	62
Issue/season :	11
Pages :	2294–2319
Peer reviewed :	Yes (verified by ORBi^lu)
Audience :	International
ISSN :	0393-0440
Abstract :	[en] We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^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 (Z_2)^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 (Z_2)^n-graded version of Liouville’s formula.
Permalink :	http://hdl.handle.net/10993/9369
DOI :	10.1016/j.geomphys.2012.07.004
Other URL :	http://www.sciencedirect.com.proxy.bnl.lu/science/article/pii/S0393044012001386
	http://arxiv.org/abs/1109.5877

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