Gerstenhaber algebra

About: Gerstenhaber algebra is a research topic. Over the lifetime, 155 publications have been published within this topic receiving 4993 citations.

Deformation Quantization of Poisson Manifolds

Abstract: I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

From Poisson algebras to Gerstenhaber algebras

Abstract: © Annales de l’institut Fourier, 1996, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

New perspectives on the BRST-algebraic structure of string theory

Abstract: Motivated by the descent equation in string theory, we give a new interpretation for the action of the symmetry charges on the BRST cohomology in terms of what we callthe Gerstenhaber bracket. This bracket is compatible with the graded commutative product in cohomology, and hence gives rise to a new class of examples of what mathematicians call aGerstenhaber algebra. The latter structure was first discussed in the context of Hochschild cohomology theory [11]. Off-shell in the (chiral) BRST complex, all the identities of a Gerstenhaber algebra hold up to homotopy. Applying our theory to thec=1 model, we give a precise conceptual description of the BRST-Gerstenhaber algebra of this model. We are led to a direct connection between the bracket structure here and the anti-bracket formalism in BV theory [29]. We then discuss the bracket in string backgrounds with both the left and the right movers. We suggest that the homotopy Lie algebra arising from our Gerstenhaber bracket is closely related to the HLA recently constructed by Witten-Zwiebach. Finally, we show that our constructions generalize to any topological conformal field theory.

Exact Gerstenhaber Algebras and Lie Bialgebroids

Abstract: We show that to any Poisson manifold and, more generally, to any triangular Lie bialgebroid in the sense of Mackenzie and Xu, there correspond two differential Gerstenhaber algebras in duality, one of which is canonically equipped with an operator generating the graded Lie algebra bracket, i.e. with the structure of a Batalin-Vilkovisky algebra.

Homotopy Gerstenhaber algebras

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 ∞-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 ∞-algebra, another, called a homotopy G-algebra, is a particular case of a B ∞-algebra, the others, a G ∞-algebra, an Ē1-algebra, and a weak G ∞-algebra, arise from the geometry of configuration spaces. Corrections to the paper of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made.