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Computing equivariant matrices on homogeneous spaces for Geometric Deep Learning and Automorphic Lie Algebras

Vincent Knibbeler

TL;DR

The paper provides an elementary, geometry-driven method to compute spaces of $G$-equivariant maps $X\to V$ for homogeneous spaces $X=G/H$, and extends this framework to invariant sections of homogeneous vector bundles and to algebra-valued fibers that yield automorphic algebras. Central to the approach is reducing the potentially infinite-dimensional problem to finite-dimensional $H$-invariants via a base point map $f:X\to G$ with $f(x)x_0=x$, establishing identifications $M_G(G/H,V)\cong V^H$ through geometric Frobenius reciprocity. When the fibers carry an algebra structure, the resulting automorphic algebras $ rak A(G,H,\frak a)$ become isomorphic to $\frak a^H$, enabling explicit classifications and connections to automorphic Lie algebras in both compact and noncompact settings. The framework applies to noncompact groups and yields practical tools for designing equivariant kernels in geometric deep learning, while also offering a pathway to study automorphic Lie algebras in novel geometries such as hyperbolic spaces and Lorentzian settings. Overall, the work unifies representation-theoretic reduction with geometric constructions to provide tractable, algebra-preserving methods for equivariance on homogeneous spaces.

Abstract

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.

Computing equivariant matrices on homogeneous spaces for Geometric Deep Learning and Automorphic Lie Algebras

TL;DR

The paper provides an elementary, geometry-driven method to compute spaces of -equivariant maps for homogeneous spaces , and extends this framework to invariant sections of homogeneous vector bundles and to algebra-valued fibers that yield automorphic algebras. Central to the approach is reducing the potentially infinite-dimensional problem to finite-dimensional -invariants via a base point map with , establishing identifications through geometric Frobenius reciprocity. When the fibers carry an algebra structure, the resulting automorphic algebras become isomorphic to , enabling explicit classifications and connections to automorphic Lie algebras in both compact and noncompact settings. The framework applies to noncompact groups and yields practical tools for designing equivariant kernels in geometric deep learning, while also offering a pathway to study automorphic Lie algebras in novel geometries such as hyperbolic spaces and Lorentzian settings. Overall, the work unifies representation-theoretic reduction with geometric constructions to provide tractable, algebra-preserving methods for equivariance on homogeneous spaces.

Abstract

We develop an elementary method to compute spaces of equivariant maps from a homogeneous space of a Lie group 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.
Paper Structure (6 sections, 8 theorems, 74 equations)

This paper contains 6 sections, 8 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Invariant sections of homogeneous vector bundles
  3. Fixed point subalgebras of homogeneous algebra bundles
  4. Discussion
  5. The Gram-Schmidt process and $\mathrm{S}^{n}\cong\mathrm{SO}(n+1)/\mathrm{SO}(n)$
  6. From homogeneous space to $\mathbb{R}^d$

Key Result

Lemma 2.3

Let $V$ be a representation of $G$ and $W$ a representation of $H$. Let $\mathbb{F}$ denote the trivial representation of $H$. Then is an isomorphism of $G$-representations if and only if $W$ is the restriction of $V$ to $H$.

Theorems & Definitions (20)

  • Example 2.1: The hyperbolic plane $\mathrm{H}^{2}$
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Example 2.6: The sphere $\mathrm{S}^{2}$
  • Corollary 2.7
  • ...and 10 more