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 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  of signs to the circumpherence . Then, the concept of associative homotopy Lie algebras admits nontrivial generalizations.