On the Transposition Anti-Involution in Real Clifford Algebras I: The Transposition Map
by Rafal Ablamowicz, Department of Mathematics TTU,
and Bertfried Fauser, School of Computer Science, The University of Birmingham, Edgbaston-Birmingham,
Tech Report No. 2010-1,
Date: May 12, 2010.
Download: downloadable file, 360KB
A particular orthogonal map on a finite dimensional real quadratic vector space (V,Q) with a non-degenerate quadratic form Q of any signature (p, q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra Cl(V*,Q) of linear functionals (multiforms) acting on the universal Clifford algebra Cl(V,Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of Cl(V,Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of Cl(V,Q). We give also an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work .
1991 Mathematics Subject Classification. 11E88, 15A66, 16S34, 20C05, 68W30.
Key words and phrases. conjugation, contraction, correlation, dual space, exterior algebra, grade involution, graded tensor product, spinor modules, indecomposable module, involution, left regular representation, minimal left ideal, monomial order, primitive idempotent, quadratic form, reversion, simple algebra, transpose of linear mapping.
Source: Email by R. Ablamowicz of 2010/06/07 23:39 (rablamowicz_at_tntech.edu)