nLab BV-algebra

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.

For details cf. Cohen & Voronov 2006, def. 5.3.1, theorem 2.1.3.

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 Cohen & Voronov 2006, theorem 5.3.3. The full homology of the framed little n-disk operad is described by Salvatore & Wahl, theorem 5.4.

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.

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:

• under good conditions, Hochschild cohomology (see there for details) carries a BV-algebra structure (eg. Felix, Thomas & Vigué-Poirrier 2004; Menichi 2009; Tradler 2008)

Related concepts

• Batalin-Vilkovisky module

• Poisson n-algebra

• homotopy BV-algebra

  • dg-BV-algebra?

• BD-algebra

References

The identification of 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, Part I in: String topology and cyclic homology, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser (2006) [math.GT/05036259, doi:10.1007/3-7643-7388-1, pdf]

• Paolo Salvatore, Nathalie Wahl: Framed discs operads and Batalin-Vilkovisky algebras, The Quarterly Journal of Mathematics 54 2 (2003) 213–231 [pdf, doi:10.1093/qmath/hag012]

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:

• Alberto Cattaneo, Domenico Fiorenza, Riccardo Longoni, Section 2 of: On the Hochschild-Kostant-Rosenberg map for graded manifolds, Int. Math. Res. Not. 2005, no. 62, 3899-3918 (arXiv:math/0503380)

and

• Claude Roger, p. 6 of: Gerstenhaber and Batalin-Vilkovisky algebras, Archivum mathematicum, Volume 45 (2009), No. 4 (pdf)

The BV-algebra structure on Hochschild cohomology:

• Yves Félix, Jean-Claude Thomas, Micheline Vigué-Poirrier: The Hochschild cohomology of a closed manifold Publ. Math. IHÉS Sci. 99 (2004) 235-252 [numdam:PMIHES_2004__99__235_0/]

• Luc Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology, Bulletin de la Société Mathématique de France, 137 2 (2009) 277-295

  [arXiv:0711.1946, numdam:BSMF_2009__137_2_277_0/, pdf]

• Thomas Tradler, The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products, Annales de l’institut Fourier 58 7 (2008) 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