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

TL;DR

This paper explores a specific orthogonal map in real Clifford algebras that induces an involutive anti-automorphism, generalizing transposition and reversion, with implications for spinor representations.

Contribution

It introduces a unique transposition anti-involution in real Clifford algebras applicable across all signatures, extending the understanding of algebra automorphisms and anti-automorphisms.

Findings

Defines a transposition anti-involution for real Clifford algebras

Shows the anti-involution reduces to reversion or conjugation in special cases

Provides an example involving real spinor spaces

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 give also an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [...II: Spabilizer groups of primitive idempotents].