nLab
BV-algebra

Definition

Definition

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

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

• equipped with a unary linear operator $\Delta :A\to A$ of degree +1

• such that

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

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

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

Definition

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

See (CohenVoronov, def. 5.3.1) 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 operad for $(n+1)$-BV-algebras is the homology of the framed little n-disk operad.

This appears as (CohenVoronov, theorem 5.3.3).

Examples

• On a smooth manifold $X$ with a volume form $\omega$ the operation of contraction with the volume form gives an isomorphism between differential forms and multivector fields. Under this isomorphism the de Rham differential on forms becomes a BV-operator on multivector fields. See there for more details.

Multivecotr field can be identified with Hochschild cohomology in good cases. So the next example is a generalization of the previous one.

• Hochschild cohomology (see there for details) carries a canonical BV-algebra structure.

Related concepts

• BV-algebra

• Poisson n-algebra

• homotopy BV-algebra

  • dg-BV-algebra?

References

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

• Ezra Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories , Comm. Math. Phys. 159 (1994), no. 2, 265–285. (arXiv)

The generalization to higher dimensional framed little disks is discussed in

• Ralph Cohen, Alexander Voronov, Notes on string topology (arXiv:math/0503625)

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

• Alberto Cattaneo, Domenico Fiorenza, Riccardo Longoni, On the Hochschild-Kostant-Rosenberg map for graded manifolds (pdf)

and on p. 6 of

• Claude Roger, Gerstenhaber and Batalin-Vilkovisky algebras (ps)

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

• Y. Félix, J.-C. Thomas, M. Vigué-Poirrier, The Hochschild cohomology of a closed manifold Publ. Math. IHÉS Sci. (2004) no 99, 235-252

• Luc Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology (pdf)