Cohomology of Yang–Mills algebras

Abstract

In this paper we compute cyclic and Hochschild homology of the universal envelope $U(YM)$ of the Yang–Mills Lie algebra $YM$. We also compute Hochschild cohomology with coefficients in $U(YM)$, considered as a bimodule over itself.

The result of the calculations depends on the number of generators $n$ of $YM$. The semidirect product $\mathfrak{so}(n)⋉C^n$ acts by derivations upon $U(YM)$. One of the important consequences of our results is that if $n\ge 3$ then the Lie algebra of outer derivations of $U(YM)$ coincides with $\mathfrak{so}(n)⋉C^n$.