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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
equipped with a unary linear operator of the same degree as the bracket
is a derivation for ;
is the failure of being a derivation for :
A -BV algebra is a similar structure with a BV-operator being of degree if is odd, and of degree if it is even.
See (Cohen-Voronov, def. 5.3.1, theorem 2.1.3) for details.
This is due to (Getzler)
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 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
Revised on December 21, 2016 16:24:29
by Urs Schreiber