On inner automorphisms preserving subspaces of Clifford algebras

TL;DR

This paper investigates inner automorphisms of Clifford algebras that preserve specific subspaces, identifying associated groups and Lie algebras, and relating them to known algebraic structures like Clifford and spin groups.

Contribution

It introduces new groups of automorphisms preserving subspaces in Clifford algebras and explores their properties and Lie algebra structures, extending to more general algebraic contexts.

Findings

Identified groups of automorphisms preserving subspaces

Characterized their Lie algebras

Connected these groups to Clifford, Lipschitz, and spin groups

Abstract

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras -- subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras -- graded central simple algebras or graded central simple algebras with involution.