Some remarks on representations of Yang-Mills algebras

TL;DR

This paper explores the representation properties of Yang-Mills algebras, revealing their ability to encompass various Lie algebras as quotients and discussing implications for solutions to Yang-Mills equations.

Contribution

It demonstrates that Yang-Mills algebras can generate a wide class of Lie algebras as quotients, extending understanding of their representation theory and applications.

Findings

Any free Lie algebra with m generators is a quotient of ym(n) for n ≥ 2m.

Any semisimple or affine Kac-Moody algebra is a quotient of ym(n) for n ≥ 4.

The quotient property fails for n=3, as ym(3) maps to sl(2,k) with solvable image.

Abstract

In this article we present some probably unexpected (in our opinion) properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra ym(n) on n generators, for n greater than or equal to 2m. We derive from this that any semisimple Lie algebra, and even any affine Kac-Moody algebra is a quotient of ym(n), for n greater than or equal to 4. Combining this with previous results on representations of Yang-Mills algebras given in an article by the author together with A. Solotar, one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from ym(3) to sl(2,k) has in fact solvable image.