A Infinity AlgebraEdit

A Infinity Algebra, more commonly written as A∞-algebra, is a sophisticated generalization of associative algebras designed to encode associativity up to a hierarchy of higher homotopies. Originating in the study of spaces and their loop structures, these algebras have since become a central tool in modern algebra, topology, and mathematical physics. They capture phenomena where products fail to be strictly associative, but do so in a controlled, coherent way that remains tractable for computation and theory.

This article surveys the core ideas, constructions, and applications of A∞-algebras, highlighting how higher operations interact, how these structures arise in practice, and where they sit in the broader landscape of homological algebra and category theory.

Definition

An A∞-algebra consists of: - a graded vector space A over a field (often of characteristic 0), viewed as a module with a grading A = ⊕ A^k, - a family of multilinear maps m_n: A^{⊗ n} → A of degree 2 − n for each integer n ≥ 1.

The maps m_n are required to satisfy a collection of coherence relations known as the Stasheff identities, which encode associativity up to higher homotopies. In practice, the first few identities illustrate the idea: - m_1 ∘ m_1 = 0, so m_1 is a differential; - m_1 commutes with m_2 in a way that makes m_2 associative up to homotopy given by m_3; - higher m_n encode higher levels of associativity relations.

A strict or unital A∞-algebra may impose additional conditions (such as m_n = 0 for n sufficiently large or the existence of a strict unit), but the general framework allows all higher maps to be nonzero when needed. The degree conventions ensure that all identities are of total degree 1 when written as a sum of terms involving the m_n.

For readability, one often writes m_1 for the differential, m_2 as an initial product, and m_n (n ≥ 3) as higher homotopies that correct associativity. The term A∞-algebra is sometimes expanded as “A-infinity algebra,” reflecting alternative naming in the literature. See A∞-algebra for the canonical formulation and variants.

Origins and historical background

The concept arose from the idea of homotopy associativity in the study of loop spaces, initiated by J. Stasheff in the 1960s in his work on A∞-spaces. The same ideas were later transported to the algebraic setting, where one replaces topological spaces with algebraic objects like chain complexes and graded vector spaces. The passage from strict associative structures to A∞-type structures provides a natural framework for working with objects up to homotopy, keeping track of higher-order corrections that matter in flexible contexts such as deformation theory and mirror symmetry. See Stasheff for the foundational results and historical development.

Structure and basic properties

Underlying data: A graded vector space A together with maps m_n: A^{⊗ n} → A of degree 2 − n.
First two operations: m_1 is a differential (m_1 ∘ m_1 = 0), and m_2 gives a product that is associative up to homotopy controlled by m_3.
Higher operations: For n ≥ 3, the maps m_n encode higher coherences that ensure a coherent, homotopy-invariant notion of multiplication.
Specializations: If m_n = 0 for all n ≥ 3, the structure reduces to a differential graded algebra (DGA) with m_1 as differential and m_2 as the product. Conversely, many algebraic constructions begin with a DGA and extend to an A∞-structure on its cohomology.
Unitality: A∞-algebras can be equipped with a unit that behaves coherently with the higher products, leading to the notion of a unital A∞-algebra.
Morphisms: An A∞-morphism f between A∞-algebras A and B is a collection of maps f_n: A^{⊗ n} → B of degree 1 − n that satisfy compatibility relations with the m_n. Quasi-isomorphisms are A∞-morphisms whose first component f_1 induces an isomorphism on cohomology.
Bar construction and coalgebras: The bar construction B(A) turns the A∞-structure into a coderivation on a cofree coalgebra, providing a powerful, functorial tool for constructing and analyzing A∞-algebras. See Bar construction for the coalgebraic perspective.
Minimal models: A central result is that any A∞-algebra is quasi-isomorphic to a minimal A∞-algebra on its cohomology, where m_1 = 0. This is known as a version of the Kadeishvili theorem and is a cornerstone of deformation theory. See Kadeishvili and minimal model for related discussions.

Morphisms, equivalences, and deformation

A∞-morphisms generalize algebra homomorphisms to higher operations, preserving the higher products in a coherent way. They enable the comparison of A∞-algebras up to homotopy and the transfer of structure along homotopy equivalences.
Quasi-isomorphisms play the role of weak equivalences in this setting, aligning with the homotopical viewpoint that focuses on invariants up to deformation.
Deformation theory: A∞-structures are central to deformation problems, where one studies how an algebraic structure deforms in a family. The Maurer–Cartan equation in an A∞-context governs deformation parameters and equivalence classes of deformations. See deformation theory and Maurer-Cartan for related concepts.
Bar and cobar constructions provide adjunctions that connect A∞-algebras with coalgebras and algebras, enabling a homotopy-theoretic mindset that is often more flexible than strict algebraic methods. See Bar construction and Cobar construction for dual points of view.

Examples and connections

DG-algebras as a special case: If all m_n vanish for n ≥ 3, an A∞-algebra reduces to a differential graded algebra (DGA) with differential m_1 and product m_2. This provides a bridge between the classical theory and the higher-homotopy world. See Differential graded algebra.
Cohomology with higher products: The cohomology H^(A) of an A∞-algebra A inherits an A∞-structure, with m_1 = 0, and with m_2 induced from the original product. Higher m_n on the cohomology capture essential extension data, making H^(A) a fertile object of study in homotopical algebra.
A∞-categories: Generalizing algebras, A∞-categories replace single objects with collections of objects and allow morphism spaces to carry A∞-structures. The most famous instance is the Fukaya category, which encodes Lagrangian submanifolds and their intersections in a symplectic manifold.
Connections to physics: In open string field theory, algebraic structures of the open sector often realize A∞-algebra (and related L∞-algebra) structures, tying higher algebra to the algebraic underpinnings of physical theories. See open string field theory and string field theory for broader context.

Applications in mathematics and physics

Homological algebra and derived categories: A∞-structures provide robust frameworks for working with derived and triangulated categories, where morphisms are understood up to higher homotopies.
Mirror symmetry: The language of A∞-algebras and A∞-categories underpins homological mirror symmetry, where one side of a dual pair is described by a Fukaya category and the other by a derived category of coherent sheaves. See homological mirror symmetry.
Deformation quantization and noncommutative geometry: Higher products encode deformation data that are central to formal deformation quantization and related noncommutative structures.
Topological field theory and invariants: A∞-structures appear in various guises when constructing invariants of manifolds and in organizing algebraic data that arise from topological field theories.

Controversies and debates

Abstraction vs. computability: A∞-algebras are highly flexible and conceptually powerful, but their full generality can obscure concrete computations. Some mathematicians favor working with strict DG-algebras when possible, reserving A∞-structures for cases where higher homotopies are genuinely needed.
Conventions and sign bookkeeping: The coherent system of signs required to satisfy the Stasheff identities can be subtle, and different sign conventions exist in the literature. This can complicate communication and cross-reference between sources, though the underlying ideas remain consistent.
Foundations and pedagogy: There is ongoing discussion about how best to introduce higher algebra to newcomers. Critics sometimes argue that early exposure to A∞-techniques can be intellectually demanding, while proponents emphasize the payoff in terms of conceptual clarity and broad applicability once the framework is absorbed.
Model-independence vs. explicit models: While the minimal-model viewpoint clarifies the existence of homotopy-invariant data, some practitioners worry that working with abstract A∞-structures can detach attention from explicit, computable models. The balance between explicit constructions and abstract existence results is a recurring theme in the field.