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BV-algebra

Contents

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 degree +1

  • 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 (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 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

Multivecotr field can be identified with Hochschild cohomology in good cases. 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

  • 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

  • Paolo Salvatore, Nathalie Wahl, Framed discs operads and Batalin- Vilkovisky algebras (pdf)

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

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)

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

Revised on October 14, 2014 17:41:21 by Zoran Škoda (161.53.130.104)