# nLab BV-algebra

Contents

### Context

#### Higher algebra

higher algebra

universal 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 $(A, \cdot, [-,-])$

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

• such that

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

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

$[-,-] = \Delta \circ (-\cdot -) - (\Delta(-) \cdot - ) - (- \cdot \Delta(-)) \,.$
###### 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 (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 $(n+1)$-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.

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

There are examples coming from Lagrangian intersection theory

• Vladimir Baranovsky, Victor Ginzburg, Gerstenhaber-Batalin-Vilkoviski structures on coisotropic intersections, arxiv/0907.0037

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

• Claude Roger, Gerstenhaber and Batalin-Vilkovisky algebras , Archivum mathematicum, Volume 45 (2009), No. 4 (pdf)

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)

• Thomas Tradler, The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products, Annales de l’institut Fourier (2008) Volume: 58, Issue: 7, page 2351-2379 (arXiv:math/0210150)

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

• Johannes Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras and

Batalin-Vilkovisky algebras_, Annales de l’institut Fourier 48:2 (1998) 425-440 eudml