Gerstenhaber algebras

Gerstenhaber algebras
Name	Gerstenhaber algebras
Type	Algebraic structure
Introduced by	Murray Gerstenhaber
Introduced date	1963
Field	Algebra, Homological algebra

Gerstenhaber algebras are graded algebraic structures introduced in work by Murray Gerstenhaber that combine a graded commutative product with a graded Lie bracket satisfying compatibility conditions. They appear naturally in the study of Hochschild cohomology, deformation theory, and mathematical physics, connecting with concepts from Murray Gerstenhaber's research to developments involving Maxim Kontsevich, Pierre Deligne, and Getzler's work on operads. These algebras serve as a bridge between structures studied by Jean-Louis Loday, Victor Ginzburg, Alexander Grothendieck, and later researchers in homological algebra and algebraic topology.

Definition

A Gerstenhaber algebra is a graded vector space V = ⊕_{i} V^i equipped with a graded commutative associative product (cup product) of degree 0 and a graded Lie bracket of degree −1. The definition axiomatizes properties observed in Hochschild cohomology of associative algebras studied by Murray Gerstenhaber and later formalized by authors influenced by Pierre Deligne's conjectures and work of Maxim Kontsevich. The bracket satisfies graded antisymmetry and the graded Jacobi identity, while the product is associative and graded commutative; compatibility is expressed by the bracket being a graded derivation with respect to the product (the Leibniz rule). Variants and extensions of the definition have been considered in research connected to Alexander Grothendieck's ideas, Victor Ginzburg's studies, and operadic approaches linked to Jim Stasheff and John Milnor.

Examples

Classical examples arise from cohomological constructions: the Hochschild cohomology HH^*(A,A) of an associative algebra A carries a Gerstenhaber structure observed by Murray Gerstenhaber and further explored by Gerald Hochschild's collaborators. The polyvector fields on a smooth manifold M produce an example through the Schouten–Nijenhuis bracket, studied by authors influenced by Henri Cartan, Élie Cartan, and later by Maxim Kontsevich in deformation quantization. In algebraic geometry, the Ext-algebra Ext^*(O_X,O_X) for a variety X relates to structures examined by Alexander Grothendieck and Jean-Pierre Serre. Examples also appear in the homology of loop spaces investigated by J. Peter May and Graeme Segal, and in string topology work by Moira Chas and Dennis Sullivan.

Properties and structure

A Gerstenhaber algebra's graded Lie bracket has degree −1, so the bracket of homogeneous elements of degrees p and q lies in degree p+q−1; this degree shift aligns with conventions used by Murray Gerstenhaber and in later expositions by Maxim Kontsevich and Pierre Deligne. The Leibniz rule relates the bracket and product, mirroring derivation properties seen in classical Lie derivative constructions associated with Élie Cartan and Henri Cartan. Universal constructions and enveloping-type constructions have analogues in literature related to Jean-Louis Loday and Gerald Hochschild, while Koszul duality perspectives connect to work by Vladimir Drinfeld and Jean-Pierre Serre. Homotopy versions (G-infinity structures) generalize classical properties and have been developed in contexts influenced by Jim Stasheff, Bernard L. Feigin, and Dennis Sullivan.

Cohomology and deformation theory

Gerstenhaber algebras play a central role in deformation theory starting from Murray Gerstenhaber's deformation theory of associative algebras and continuing through Maxim Kontsevich's formality theorem. The Hochschild cohomology HH^*(A,A) carries both the cup product and the Gerstenhaber bracket, and governs infinitesimal deformations of associative algebras, a theme connected to concepts advanced by Alexander Grothendieck and Pierre Deligne. The compatibility conditions enable the description of obstruction classes and control of deformation problems, as elaborated in expositions influenced by Michel Demazure and Grothendieck-style cohomological methods. Formality results linking polyvector fields and Hochschild cochains, notably by Maxim Kontsevich, exploit the Gerstenhaber structure to construct L-infinity quasi-isomorphisms and to resolve deformation quantization problems that attracted interest from researchers like Alain Connes and André Lichnerowicz.

Relations to other algebraic structures

Gerstenhaber algebras relate closely to Batalin–Vilkovisky algebras studied by I. A. Batalin and G. A. Vilkovisky, with BV operators providing generators for the Gerstenhaber bracket in many geometric examples influenced by Dennis Sullivan and Moira Chas. Operadic formulations tie Gerstenhaber structures to the homology of the little disks operad, a connection developed in work by Fred Cohen, Vladimir Drinfeld, and Maxim Kontsevich, and elaborated upon by Murray Gerstenhaber's successors such as Getzler and John McClure. Homotopy Gerstenhaber (G-infinity) algebras generalize to A-infinity and L-infinity frameworks introduced by Jim Stasheff and Michèle Vergne, linking to associative and Lie homotopy theories studied by Gerald Hochschild and Jean-Louis Loday.

Applications and occurrences

Gerstenhaber algebras appear in deformation quantization of Poisson manifolds, central to Maxim Kontsevich's work, and in string topology constructions developed by Moira Chas and Dennis Sullivan. They arise in the study of algebraic structures on Ext and Tor groups in representation theory contexts involving institutions like École Normale Supérieure and research by Alexander Grothendieck-influenced schools. In mathematical physics, Gerstenhaber-type structures inform the BV formalism employed in studies by I. A. Batalin and G. A. Vilkovisky and influence quantum field theoretic approaches pursued at institutions such as Institut des Hautes Études Scientifiques and Perimeter Institute. Further occurrences include applications to deformation problems in noncommutative geometry explored by Alain Connes and to operadic and homotopical algebra advanced by Vladimir Drinfeld and Jim Stasheff.

Category:Algebraic structures