On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras
E. R. Filimoshina, D. S. Shirokov
TL;DR
The paper addresses the problem of classifying Lie groups that preserve subspaces of degenerate geometric algebras under adjoint and twisted adjoint representations. It develops a norm-function framework using $\psi(T)=\widetilde{T}T$ and $\chi(T)=\widehat{\widetilde{T}}T$ to define generalized Clifford/Lipschitz groups $Q^{\overline{k}}$ and $\check{Q}^{\overline{k}}$, and establishes precise equivalences with the groups $\Gamma^{\overline{k}}$, $\check{\Gamma}^{\overline{k}}$, and $\tilde{\Gamma}^{\overline{k}}$ for various $k$ under the representations ${\rm ad}$, ${\check{\rm ad}}$, and ${\tilde{\rm ad}}$. The work provides explicit centralizer structures, concrete low-dimensional examples, and a detailed Lie-algebra correspondence linking these groups to familiar algebras such as $su(2)$, $sl(2,\mathbb{C})$, Heisenberg algebras, and the Poincaré algebra. By demonstrating connections to matrix groups (e.g., ${\rm UT}(2,\mathbb{F})$, ${\rm UT}(4,\mathbb{F})$) and the Heisenberg group ${\rm Heis}_4$, the results illuminate both the algebraic structure and potential computational applications. The results are poised to influence constructions of generalized spin groups and equivariant mappings in areas like physics and equivariant neural networks. Overall, the paper extends Clifford/Lipschitz group theory to degenerate geometric algebras, enabling systematic analysis of symmetry-preserving transformations across a broad class of algebras.
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.