Discovering Sparse Representations of Lie Groups with Machine Learning

TL;DR

This paper introduces a machine learning method to discover sparse representations of Lie groups, successfully reproducing known generators and applicable to any Lie group, advancing symmetry analysis in mathematical physics.

Contribution

The paper presents a novel machine learning approach for deriving sparse Lie algebra representations, extending previous symmetry transformation techniques to arbitrary Lie groups.

Findings

Reproduces canonical generators of Lorentz, U(n), and SU(n) groups

Demonstrates general applicability to any Lie group

Provides a new tool for symmetry analysis in physics

Abstract

Recent work has used deep learning to derive symmetry transformations, which preserve conserved quantities, and to obtain the corresponding algebras of generators. In this letter, we extend this technique to derive sparse representations of arbitrary Lie algebras. We show that our method reproduces the canonical (sparse) representations of the generators of the Lorentz group, as well as the $U(n)$ and $SU(n)$ families of Lie groups. This approach is completely general and can be used to find the infinitesimal generators for any Lie group.