symmetric monoidal (∞,1)-category of spectra
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 $(A, \cdot, [-,-])$
equipped with a unary linear operator $\Delta : A \to A$ of the same degree as the bracket
such that
$\Delta$ is a derivation for $[-,-]$;
$[-,-]$ is the failure of $\Delta$ being a derivation for $\cdot$:
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.
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 $(n+1)$-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
The generalization to higher dimensional framed little disks is discussed in
Ralph Cohen, Alexander Voronov, Notes on string topology (arXiv:math/0503625)
Paolo Salvatore, Nathalie Wahl?, Framed discs operads and Batalin-Vilkovisky algebras (pdf)
There are examples coming from Lagrangian intersection theory:
The BV-algebra structure on multivector fields on an oriented smooth manifold is discussed for instance in:
and
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
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:
Last revised on October 14, 2022 at 21:13:20. See the history of this page for a list of all contributions to it.