On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras
This work provides a theoretical foundation for applications in physics and computer science, such as equivariant neural networks, but appears incremental as it extends known concepts in geometric algebras.
The paper introduces Lie groups in degenerate Clifford algebras that preserve four subspaces defined by involutions and representations, proving their equivalence to norm functions in spin group theory and studying their Lie algebras, with some related to Heisenberg structures.
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.