Some remarks on representations of Yang-Mills algebras

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 ≥ 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 ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [4], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 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.