On some generalizations of Batalin-Vilkovisky algebras (Q1368587)

A Batalin-Vilkovisky algebra (BVA) is a supercommutative associative algebra \({\mathcal A}\) with an odd, square zero, second order differential operator \(\Delta\). The author introduces a definition of higher-order differential operators on a superalgebra. One of the justifications given by the author is that the nonnegative components of a vertex operator in a vertex operator superalgebra are such differential operators with respect to the algebraic operation defined by the \(-1\)-component of the vertex operators. This fact leads the author to introduce the notions of generalized Batalin-Vilkovisky algebras (GBVAs) and vertex operator Batalin-Vilkovisky algebras (VOBVAs). It is proved that a GBVA satisfies the modified skew-symmetry, Leibniz rule, Poisson rule and derivation rule for \(\Delta\). Some further properties and examples of GBVAs and VOBVAs are also discussed.