Computing equivariant matrices on homogeneous spaces for Geometric Deep Learning and Automorphic Lie Algebras
This work provides theoretical tools for geometric deep learning and automorphic Lie algebras, but it appears incremental as it builds on existing mathematical frameworks.
The paper tackles the problem of computing spaces of equivariant maps from homogeneous spaces to group modules, developing an elementary method that does not require compactness of the Lie group, and classifies automorphic algebras for cases with compact stabilizers.
We develop an elementary method to compute spaces of equivariant maps from a homogeneous space $G/H$ of a Lie group $G$ to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras.