Generalized Degenerate Clifford and Lipschitz Groups in Geometric Algebras

TL;DR

This paper defines and analyzes generalized degenerate Clifford and Lipschitz groups within geometric algebras, exploring their algebraic properties, relations, and potential applications in various scientific fields.

Contribution

It introduces a new class of Lie groups in geometric algebras, characterizes them via centralizers, and examines their Lie algebras and specific cases, expanding the theoretical framework.

Findings

Defined generalized degenerate Clifford and Lipschitz groups.

Established their relations with centralizers and norm functions.

Explored applications in physics, computer science, and engineering.

Abstract

This paper introduces and studies generalized degenerate Clifford and Lipschitz groups in geometric (Clifford) algebras. These Lie groups preserve the direct sums of the subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations in degenerate geometric algebras. We prove that the generalized degenerate Clifford and Lipschitz groups can be defined using centralizers and twisted centralizers of fixed grades subspaces and the norm functions that are widely used in the theory of spin groups. We study the relations between these groups and consider them in the particular cases of plane-based geometric algebras and Grassmann algebras. The corresponding Lie algebras are studied. The presented groups are interesting for the study of generalized degenerate spin groups and applications in computer science, physics, and engineering.