We define the concept of higher order differential operators on a general noncommutative, nonassociative superalgebra A, and show that a vertex operator superalgebra (VOSA) has plenty of them, namely modes of vertex operators. A linear operator ∆ is a differential operator of order ≤ r if an inductively defined (r + 1)-linear form Φ r+1 ∆ with values in A is identically zero. These forms resemble the multilinear string products of Zwiebach. When A is a "classical" (i.e. supercommutative, associative) algebra, and ∆ is an odd, square zero, second order differential operator on A, Φ 2 ∆ defines a "Batalin-Vilkovisky algebra" structure on A. Now that a second order differential operator makes sense, we generalize this notion to any superalgebra with such an operator, and show that all properties of the classical BV bracket but one continue to hold "on the nose". As special cases, we provide several examples of classical BV algebras, vertex operator BV algebras, and differential BV algebras. We also point out connections to Leibniz algebras and the noncommutative homology theory of Loday. Taking the generalization process one step further, we remove all conditions on the odd operator ∆ to examine the changes in the basic properties of the bracket. We see that a topological chiral algebra with a mild restriction yields a classical BV algebra in the cohomology. Finally, we investigate the quantum BV master equation for (i) classical BV algebras, (ii) vertex operator BV algebras, and (iii) generalized BV algebras, relating it to deformations of differential graded algebras.

We generalize the coupled braces {x}{y} of Gerstenhaber and {x}{y1, . . . , yn} of Getzler depicting compositions of multilinear maps in the Hochschild complex C • (A) = Hom(T A; A) of a graded vector space A to expressions of the form {x, and clarify many of the existing sign conventions that show up in the algebra of mathematical physics (namely in associative and Lie algebras, Batalin-Vilkovisky algebras, A∞ and L∞ algebras). As a result, we introduce a new variant of the master identity for L∞ algebras. We also comment on the bialgebra cohomology differential of Gerstenhaber and Schack, and define multilinear higher order differential operators with respect to multilinear maps using the new language. The continuation of this work will be on the various homotopy structures on a topological vertex operator algebra, as introduced

We produce a master identity {m}{m} = 0 for homotopy Gerstenhaber algebras, as defined by Getzler and Jones and utilized by Kimura, Voronov, and Zuckerman in the context of topological conformal field theories. To this end, we introduce the notion of a "partitioned multilinear map" and explain the mechanics of composing such maps. In addition, many new examples of pre-Lie algebras and homotopy Gerstenhaber algebras are given.

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