On the associative homotopy Lie algebras and the Wronskians

Representations of the Schlessinger-Stasheff's associative homotopy Lie algebras in the spaces of higher-order differential operators are analyzed; in particular, a remarkable identity for the Wronskian determinants is obtained. The W-transformations of chiral embeddings, related with the Toda equations, of complex curves into the Kaehler manifolds are shown to be endowed with the homotopy Lie algebra structures. Extensions of the Wronskian determinants that preserve the properties of the Schlessinger-Stasheff's algebras are constructed for the case of $n\geq1$ independent variables.