nLab BV-algebra




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



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(-)) \,.

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.



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

This is due to (Getzler)


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).


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 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

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:


  • 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:

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

Last revised on October 14, 2022 at 21:13:20. See the history of this page for a list of all contributions to it.