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 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].