Isometries of Clifford algebras II

Abstract

Let F be a field of characteristic different from 2, and let $F^n$ denote the vector space of n-tuples of elements in F. Let $e_1,...,e_n$ denote the canonical basis of $F^n$. Let r and s be nonnegative integers such that r + s = n, and let Q denote the nondegenerate bilinear form on $F^n$ such that $Q(e_i,e_j)=0$ if i,j are distinct, $Q(e_i,e_i)=1$ if $1\le i\le r$ and $Q(e_{r+j},e_{r+j})=-1$ if $1\le j\le s$. Let $C\ell (r,s)$ denote the Clifford algebra determined by Q and $F^n$. There is a canonical extension of Q to a nondegenerate, symmetric, bilinear form $\bar{Q}$ on $C\ell (r,s)$. An element g of $C\ell (r,s)$ will be called an isometry of $C\ell (r,s)$ if left and right translations by g preserve $\bar{Q}$. Let $G_{r,s}$ denote the group of all isometries of $C\ell (r,s)$. We construct a Lie algebra $L{G}_{r,s}$ over F that equals the Lie algebra of $G_{r,s}$ in the case that F = R or C. The Lie algebra $L{G}_{r,s}$ admits an involutive automorphism whose +1 and -1 eigenspaces determine a Cartan decomposition $L{G}_{r,s}=K_{r,s}\oplus P_{r,s}$. We compute the bracket relations for a natural system of generators of $L{G}_{r,s}$. Finally, we determine $L{G}_{r,s}$ in the case that F = R.

arXiv comment31 pages, changes after abstract