On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras

TL;DR

This paper defines new Lie groups within degenerate Clifford algebras that preserve key subspaces, linking them to spin groups and Heisenberg groups, with applications in physics and neural networks.

Contribution

It introduces and characterizes Lie groups in degenerate Clifford algebras that preserve fundamental subspaces, connecting them to existing algebraic structures and potential applications.

Findings

Lie groups preserving subspaces are characterized using norm functions.

These groups relate to Heisenberg Lie groups and algebras.

Potential applications in physics and equivariant neural networks.

Abstract

This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that these Lie groups can be equivalently defined using norm functions of multivectors applied in the theory of spin groups. We also study the corresponding Lie algebras. Some of these Lie groups and algebras are closely related to Heisenberg Lie groups and algebras. The introduced groups are interesting for various applications in physics and computer science, in particular, for constructing equivariant neural networks.