On the transposition anti-involution in real Clifford algebras I: the transposition map

Abstract

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 also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].