On Some Lie Groups in Degenerate Clifford Geometric Algebras

TL;DR

This paper introduces five new families of Lie groups within degenerate Clifford geometric algebras, encompassing known groups like spin, Lipschitz, and Clifford groups, with potential applications across multiple scientific fields.

Contribution

It defines and analyzes five novel Lie groups in degenerate Clifford algebras, expanding the understanding of their structure and subgroups in arbitrary dimensions and signatures.

Findings

The Lie groups preserve specific subspaces under adjoint and twisted adjoint representations.

The groups include degenerate spin, Lipschitz, and Clifford groups as subgroups.

Potential applications in physics, engineering, and computer science are identified.

Abstract

In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in the case of arbitrary dimension and signature. The considered Lie groups can be of interest for various applications in physics, engineering, and computer science.