L-infinity algebra

L-infinity algebra
Name	L-infinity algebra
Field	Algebra, Category:Algebraic topology
Introduced	1970s–1990s
Authors	Stasheff, Lada, Zwiebach

L-infinity algebra is a homotopy-invariant generalization of a Lie algebra that encodes higher-order multilinear brackets satisfying generalized Jacobi identities up to coherent homotopies. It arose in interactions among algebraic topology, mathematical physics, and deformation theory and plays a central role in modern approaches to homological algebra, operad theory, and derived geometry.

Definition

An L-infinity algebra is a graded vector space equipped with a collection of graded skew-symmetric multilinear maps l_n for n ≥ 1 that satisfy a hierarchy of quadratic relations generalizing the Jacobi identity; equivalently it can be described as a codifferential on the cofree cocommutative coalgebra cogenerated by the suspension of the underlying graded space. This structure was developed following ideas from Jim Stasheff, Tomas Lada, and Barton Zwiebach and connects to concepts from Gerstenhaber algebra, BV algebra, Chevalley–Eilenberg complex, and Koszul duality in operad theory. The defining relations can be framed using the language of coderivations, coalgebras, and suspension operators employed in the work of Victor Ginzburg, Bertrand Toën, and Maxim Kontsevich.

Examples

Classic examples include ordinary Lie algebras recovered when all higher brackets vanish beyond l_2, and differential graded Lie algebras appearing in the study of Hochschild cohomology and the deformation quantization program of Kontsevich. String field theory constructions of Zwiebach provide L-infinity structures in closed string field theory and relate to models from Witten and E. Witten's work on topological quantum field theory. The space of polyvector fields on a Calabi–Yau manifold carries structures related to L-infinity algebras via the Schouten–Nijenhuis bracket and Gerstenhaber algebra techniques used by Maxim Kontsevich and Dmitry Tamarkin. Derived brackets of Kosmann-Schwarzbach produce explicit l_n on shifted complexes encountered in Poisson geometry and symplectic geometry contexts studied by Alan Weinstein and Jean-Louis Loday.

Homotopy and Minimal Models

L-infinity algebras admit minimal model theory parallel to Sullivan minimal models in rational homotopy theory; any L-infinity algebra is quasi-isomorphic to a minimal L-infinity algebra with vanishing unary bracket, providing a canonical homotopy invariant analogous to Dennis Sullivan's constructions. Homotopy transfer theorems allow one to transfer L-infinity structures along deformation retracts as in work by Kadeishvili, Merkulov, and Markl, connecting to notions in A-infinity algebra theory developed by Jim Stasheff and Bernhard Keller. Minimal models are instrumental in computations in string topology inspired by Chas–Sullivan and in comparing derived moduli problems studied by Jacob Lurie and Bertrand Toën.

Morphisms and Quasi-isomorphisms

Morphisms of L-infinity algebras are given by collections of multilinear maps intertwining the brackets up to higher homotopies; strict morphisms reduce to ordinary Lie algebra homomorphisms while general L-infinity morphisms capture homotopy equivalences relevant to derived category frameworks in the work of Bondal and Dmitri Orlov. Quasi-isomorphisms are L-infinity morphisms inducing isomorphisms on cohomology and play the role of weak equivalences in model categories used by Hinich, Getzler, and Pridham when comparing moduli stacks in derived algebraic geometry developed by Toën and Lurie.

Cohomology and Deformation Theory

The cohomology of an L-infinity algebra generalizes Chevalley–Eilenberg cohomology and controls deformation problems via Maurer–Cartan elements solving an infinite sequence of equations; the formal deformation functor associated to an L-infinity algebra is governed by classical results of Deligne, Drinfeld, and Goldman–Millson. Obstruction theory and unobstructedness criteria connect to formality theorems such as Kontsevich formality, with applications to deformation quantization on Poisson manifolds studied by Maxim Kontsevich and Yvette Kosmann-Schwarzbach. The interplay between L-infinity cohomology and Hodge theory appears in works by Barannikov and Kontsevich on moduli of complex structures.

Applications in Physics and Geometry

In theoretical physics, L-infinity algebras organize gauge symmetries and interactions in classical and quantum field theories, featuring prominently in formulations by Henneaux, Barnich, and Zwiebach and in BV–BRST frameworks originating from Batalin–Vilkovisky and Becchi–Rouet–Stora–Tyutin methods. In geometry, they appear in descriptions of derived intersections, shifted symplectic structures studied by Pantev–Toën–Vaquié–Vezzosi, and in mirror symmetry contexts investigated by Kontsevich and Strominger, Yau, and Zaslow. L-infinity techniques underpin modern approaches to moduli of complex and vector bundles, relating to work by Donaldson and Richard Thomas on enumerative invariants.

Constructions and Transfer Theorems

Standard constructions include bar and cobar constructions from homological algebra, homotopy transfer via homological perturbation lemmas developed by Gugenheim, and operadic methods using the Lie operad and its homotopy resolution by Koszul duality techniques from Getzler and Jones and Ginzburg–Kapranov. The homotopy transfer theorem ensures that L-infinity structures descend to cohomology or to deformation retracts, enabling effective computations in examples drawn from Hochschild cohomology, string topology, and symplectic field theory studied by Eliashberg, Givental, and Seidel.

Category:Algebraic structures