nLab
BV-algebra

Contents

Idea

For the actual relation to BV-complexes see at relation between BV and BD.

Definition

Definition

A Batalin-Vilkovisky algebra or BV-algebra for short is

  • a Gerstenhaber algebra (A,,[,])(A, \cdot, [-,-])

  • equipped with a unary linear operator Δ:AA\Delta : A \to A of the same degree as the bracket

  • such that

    1. Δ\Delta is a derivation for [,][-,-];

    2. [,][-,-] is the failure of Δ\Delta being a derivation for \cdot:

      [,]=Δ()(Δ())(Δ()). [-,-] = \Delta \circ (-\cdot -) - (\Delta(-) \cdot - ) - (- \cdot \Delta(-)) \,.
Definition

A (n+1)(n+1)-BV algebra is a similar structure with a BV-operator being of degree nn if nn is odd, and of degree n/2n/2 if it is even.

See (Cohen-Voronov, def. 5.3.1, theorem 2.1.3) for details.

Properties

Theorem

The operad for BV-algebras is the homology of the framed little 2-disk operad.

This is due to (Getzler)

Theorem

The homology of an algebra over an operad over the framed little n-disk operad has a natural structure of an (n+1)(n+1)-BV-algebra.

This appears as (CohenVoronov, theorem 5.3.3). The full homology of the framed little n-disk operad is described by (SalvatoreWahl, theorem 5.4).

Examples

Multivector fields may be identified with Hochschild cohomology in good cases (the Hochschild-Kostant-Rosenberg theorem). So the next example is a generalization of the previous one.

References

The identification o BV-algebras as algebras over the homology of the framed little disk operad is due to

The generalization to higher dimensional framed little disks is discussed in

There are examples coming from Lagrangian intersection theory

The BV-algebra structure on multivector fields on an oriented smooth manifold is discussed for instance in section 2 of

and on p. 6 of

The BV-algebra structure on Hochschild cohomology is discussed for instance in

There is a prominent class of examples coming from Lie-Rinehart algebras