# Isometries of Clifford Algebras I

## TL;DR

This paper investigates the structure of isometry groups and their Lie algebras for Clifford algebras over fields with characteristic not 2, providing explicit computations in the real positive definite case.

## Contribution

It derives structural properties of the Lie algebra of infinitesimal isometries and computes the isometry group for real positive definite Clifford algebras.

## Key findings

- Computed the isometry group for real positive definite Clifford algebras.
- Derived basic structural information about the Lie algebra of infinitesimal isometries.
- Set the stage for future work on nondegenerate forms over real vector spaces.

## Abstract

Let $V$ be a finite dimensional vector space over a field $F$ of characteristic different from 2, and let $Q$ be a nondegenerate, symmetric, bilinear form on $V$. Let $C\ell(V,Q)$ be the Clifford algebra determined by $V$ and $Q$. The bilinear form $Q$ extends in a natural way to a nondegenerate, symmetric, bilinear form $\bar{Q}$ on $C\ell(V,Q)$. Let $G$ be the group of isometries of $C\ell(V,Q)$ relative to $\bar{Q}$, and let $LG$ be the Lie algebra of infinitesimal isometries of $C\ell(V,Q)$ relative to $\bar{Q}$. We derive some basic structural information about $LG$, and we compute $G$ in the case that $F = R, V = R^{n}$ and $Q$ is positive definite on $R^{n}$. In a sequel to this paper we determine $LG$ in the case that $F = R, V = R^{n}$ and $Q$ is nondegenerate on $R^{n}$.