On homotopy Lie algebra structures in the rings of differential operators

Abstract

We study the Schlessinger-Stasheff's homotopy Lie structures on the associative algebras of differential operators Diff$_\ast(K^n)$ w.r.t. n independent variables.The Wronskians are proved to provide the relations for the generators of these algebras; two remarkable identities for the Wronskian and the Vandermonde determinants are obtained. We axiomize the idea of the Hochschild cohomologies and extend the group $\mathbb{Z}_2$ of signs $(-1)^\sigma$ to the circumpherence $S^1$. Then, the concept of associative homotopy Lie algebras admits nontrivial generalizations.