# Isometries of Clifford algebras II

## TL;DR

This paper studies the isometry groups of Clifford algebras over fields with characteristic not 2, constructs associated Lie algebras, and explicitly determines these structures over the real numbers.

## Contribution

It constructs a Lie algebra for the isometry group of Clifford algebras and computes its structure, extending known results to a broader algebraic setting.

## Key findings

- Constructed the Lie algebra LG_{r,s} for isometries of Clifford algebras.
- Determined the Lie algebra structure over the real numbers.
- Computed the bracket relations for generators of LG_{r,s}.

## 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 $1 \leq i \leq r$ and $Q(e_{r+j},e_{r+j}) = -1$ if $1 \leq j \leq s$. 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 $LG_{r,s}$ over F that equals the Lie algebra of $G_{r,s}$ in the case that F = R or C. The Lie algebra $LG_{r,s}$ admits an involutive automorphism whose +1 and -1 eigenspaces determine a Cartan decomposition $LG_{r,s} = K_{r,s} \oplus P_{r,s}$. We compute the bracket relations for a natural system of generators of $LG_{r,s}$. Finally, we determine $LG_{r,s}$ in the case that F = R.