Transposition anti-involution in Clifford algebras and invariance groups of scalar products on spinor spaces

Abstract

We introduce on the abstract level in real Clifford algebras \cl_{p,q} of a non-degenerate quadratic space (V,Q), where Q has signature \epsilon=(p,q), a transposition anti-involution \tp. In a spinor representation, the anti-involution \tp gives transposition, complex Hermitian conjugation or quaternionic Hermitian conjugation when the spinor space \check{S} is viewed as a \cl_{p,q}-left and \check{K}-right module with \check{K} isomorphic to R or R^2, C, or, H or H^2. \tp is a lifting to \cl_{p,q} of an orthogonal involution \tve: V \rightarrow V which depends on the signature of Q. The involution is a symmetric correlatio \tve: V \rightarrow V^{*} \cong V and it allows one to define a reciprocal basis for the dual space (V^{*},Q). The anti-involution \tp acts as reversion on \cl_{p,0} and as conjugation on \cl_{0,q}. Using the concept of a transpose of a linear mapping one can show that if [L_u] is a matrix in the left regular representation of the operator L_u: \cl_{p,q} \rightarrow \cl_{p,q} relative to a Grassmann basis B in \cl_{p,q}, then matrix [L_{\tp(u)}] is the matrix transpose of [L_u]. Of particular importance is the action of \tp on the algebraic spinor space S, generated by a primitive idempotent f, or a sum f+\hat{f} in simple or semisimple algebras. \tp allows us to define a new spinor scalar product S \times S \rightarrow \check{K}, where K=f\cl_{p,q}f and \check{K}=K or K \oplus \hat{K} in the simple or semisimple case. Our scalar product reduces to well known ones in Euclidean and anti-Euclidean signatures. \tp acts as identity, complex conjugation, or quaternionic conjugation on \check{K}. The action of \tp on spinors results in matrix transposition, complex Hermitian conjugation, or quaternionic ermitian conjugation. We classify the automorphism groups of the new product as O(N), U(N), Sp(N), O(N)^2, or Sp(N)^2.