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A Batalin-Vilkovisky algebra or BV-algebra for short is
a Gerstenhaber algebra
equipped with a unary linear operator of degree +1
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 (CohenVoronov, def. 5.3.1) 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
- 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
- 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 August 31, 2016 10:26:30
by Urs Schreiber