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Lie-Rinehart algebras, Gerstenhaber algebras, and B-V algebras  [PDF]
Johannes Huebschmann
Mathematics , 1997,
Abstract: For a Lie-Rinehart algebra (A,L), generators for the Gerstenhaber algebra \Lambda_A L correspond bijectively to right (A,L)-connections on A in such a way that B-V structures correspond to right (A,L)-module structures on A. When L is projective as an A-module, given an exact generator \partial, the homology of the B-V algebra (\Lambda_A L,\partial) coincides with that of L with coefficients in A with respect to the right (A,L)-module structure determined by \partial. When L is also of finite rank n, there are bijective correspondences between (A,L)-connections on \Lambda_A^nL and right (A,L)-connections on A and between left (A,L)- module structures on \Lambda_A^nL and right (A,L)-module structures on A. Hence there are bijective correspondences between (A,L)-connections on \Lambda_A^n L and generators for the Gerstenhaber bracket on \Lambda_A L and between (A,L)-module structures on \Lambda_A^n L and B-V algebra structures on \Lambda_A L. The homology of such a B-V algebra (\Lambda_A L,\partial) coincides with the cohomology of L with coefficients in \Lambda_A^n L, for the left (A,L)-module structure determined by \partial. Some applications are discussed.
Multi derivation Maurer-Cartan algebras and sh-Lie-Rinehart algebras  [PDF]
Johannes Huebschmann
Mathematics , 2013,
Abstract: We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we first develop a characterization of sh Lie-Rinehart algebras in terms of differential graded cocommutative coalgebras and Lie algebra twisting cochains that extends the nowadays standard characterization of an ordinary sh Lie algebra (equivalently: Linfty algebra) in terms of its associated generalized Cartan-Chevalley-Eilenberg coalgebra. Our approach avoids any higher brackets but reproduces these brackets in a conceptual manner. The new technical tool we develop is a notion of filtered multi derivation chain algebra, somewhat more general than the standard notion of a multicomplex endowed with a compatible algebra structure. The crucial observation, just as for ordinary Lie-Rinehart algebras, is this: For a general sh Lie-Rinehart algebra,the generalized Cartan-Chevalley-Eilenberg operator on the corresponding graded algebra involves two operators, one coming from the sh Lie algebra structure and the other from the generalized action on the corresponding algebra; the sum of the operators is defined on the algebra while the operators are individually defined only on a larger ambient algebra. We illustrate the structure with quasi Lie-Rinehart algebras.
Gerstenhaber brackets on Hochschild cohomology of quantum symmetric algebras and their group extensions  [PDF]
Sarah Witherspoon,Guodong Zhou
Mathematics , 2014,
Abstract: We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute brackets also in some cases for skew group algebras arising as group extensions of quantum symmetric algebras.
Rinehart complexes and Batalin-Vilkovisky algebras  [PDF]
Johannes Huebschmann
Mathematics , 2000,
Abstract: For a Lie-Rinehart algebra (A,L) such that, as an A-module, L is finitely generated and projective of finite constant rank, the relationship between generators of the Gerstenhaber bracket and connections on the highest A-exterior power of L given in an earlier paper arises from the canonical pairing between the exterior A-powers of L. Thus, given an exact generator for the corresponding Gerstenhaber algebra, the chain complex underlying the resulting Batalin-Vilkovisky algebra coincides with the Rinehart complex computing the corresponding Lie-Rinehart homology.
Homotopy Gerstenhaber Structures and Vertex Algebras  [PDF]
Imma Gálvez,Vassily Gorbounov,Andrew Tonks
Mathematics , 2006, DOI: 10.1007/s10485-008-9170-3
Abstract: We provide a simple construction of a Gerstenhaber-infinity algebra structure on a class of vertex algebras V, which lifts the strict Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications: the construction of a sheaf of Gerstenhaber-infinity algebras on a Calabi-Yau manifold extending the multiplication and bracket of functions and vector fields, and of a Lie-infinity structure related to the bracket of Courant.
Duality in Gerstenhaber algebras  [PDF]
Yves Félix,Luc Menichi,Jean-Claude Thomas
Mathematics , 2002,
Abstract: Let $C$ be a differential graded coalgebra, $ \bar\Omega C$ the Adams cobar construction and $C^\vee$ the dual algebra. We prove that for a large class of coalgebras $C$ there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies $HH^\ast (C^\vee, C ^\vee)$ and $HH^\ast (\bar\Omega C ; \bar\Omega C)$. This result permits to describe a Hodge decomposition of the loop space homology of a closed oriented manifold, in the sense of Chas-Sullivan, when the field of coefficients is of characteristic zero.
Homotopy Gerstenhaber algebras  [PDF]
Alexander A. Voronov
Mathematics , 1999,
Abstract: The purpose of this paper is to complete Getzler-Jones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the B_infty-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: one of them is a B_infty-algebra, another, called a homotopy G-algebra, is a particular case of a B_infty-algebra, the others, a G_infty-algebra, an E^1-bar-algebra, and a weak G_infty-algebra, arise from the geometry of configuration spaces. Corrections to the paper math.QA/9602009 of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made.
Tensor products of homotopy Gerstenhaber algebras  [PDF]
Matthias Franz
Mathematics , 2010, DOI: 10.4310/HHA.2011.v13.n2.a15
Abstract: On the tensor product of two homotopy Gerstenhaber algebras we construct a Hirsch algebra structure which extends the canonical dg algebra structure. Our result applies more generally to tensor products of "level 3 Hirsch algebras" and also to the Mayer-Vietoris double complex.
Universal central extensions of Lie-Rinehart algebras  [PDF]
José Luis Castiglioni,Xabier García-Martínez,Manuel Ladra
Mathematics , 2014,
Abstract: In this paper we study the universal central extension of a Lie--Rinehart algebra and we give a description of it. Then we study the lifting of automorphisms and derivations to central extensions. We also give a definition of a non-abelian tensor product in Lie--Rinehart algebras based on the construction of Ellis of non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product to the universal central extension.
Representations of Homotopy Lie-Rinehart Algebras  [PDF]
Luca Vitagliano
Mathematics , 2013, DOI: 10.1017/S0305004114000541
Abstract: I propose a definition of left/right connection along a strong homotopy Lie-Rinehart algebra. This allows me to generalize simultaneously representations up to homotopy of Lie algebroids and actions of strong homotopy Lie algebras on graded manifolds. I also discuss the Schouten-Nijenhuis calculus associated to strong homotopy Lie-Rinehart connections.
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