nLab
A-infinity-algebra

Contents

Idea

An A A_\infty-algebra is a monoid internal to a homotopical category such that the associativity law holds not as an equation, but only up to higher coherent homotopy.

Definition

Definition

An A A_\infty-algebra is an algebra over an operad over an A-∞ operad.

Realizations

In chain complexes

Let here \mathcal{E} be the category of chain complexes 𝒞𝒽 \mathcal{Ch}_\bullet. Notice that often in the literature this choice of \mathcal{E} is regarded as default and silently assumed.

An A A_\infty-algebra in chain complexes is concretely the following data.

A chain A A_\infty-algebra is the structure of a degree 1 coderivation

D:T cVT cV D : T^c V \to T^c V

on the reduced tensor coalgebra? T cV= n1V nT^c V = \oplus_{n\geq 1} V^{\otimes n} over a graded vector space VV (the coproduct is the unshuffle product) such that

D 2=0. D^2 = 0 \,.

Coderivations on free coalgebras are entirely determined by their “value on cogenerators”, which allows to decompose

D=D 1+D 2+D 3+ D = D_1 + D_2 + D_3 + \cdots

with each D kD_k specified entirely by its action

D k:V kV. D_k : V^{\otimes k} \to V \,.

which is a map of degree 2k2-k (or can be alternatively understood as a map D k:(V[1]) kV[1]D_k : (V[1])^{\otimes k}\to V[1] of degree 11).

Then:

  • D 1:VVD_1 : V\to V is the differential with D 1 2=0D_1^2 = 0;

  • D 2:V 2VD_2 : V^{\otimes 2} \to V is the product in the algebra;

  • D 3:V 3VD_3 : V^{\otimes 3} \to V is the associator which measures the failure of D 2D_2 to be associative;

  • D 4:V 4VD_4 : V^{\otimes 4} \to V is the pentagonator (or so) which measures the failure of D 3D_3 to satisfy the pentagon identity;

  • and so on.

One can also allow D 0D_0, in which case one talks about weak A A_\infty-algebras, which are less understood.

There is a resolution of the operad Ass\mathrm{Ass} of associative algebras (as operad on the category of chain complexes) which is called the A A_\infty-operad; the algebras over the A A_\infty-operad are precisely the A A_\infty-algebras.

A morphism of A A_\infty-algebras f:ABf : A\to B is a collection {f n} n1\lbrace f_n\rbrace_{n\geq 1} of maps

f n:(A[1]) nB[1] f_n : (A[1])^{\otimes n}\to B[1]

of degree 00 satisfying

0i+jnf i+j+1(1 iD nij1 j)= i 1++i r=nD r(f i 1f i r). \sum_{0\leq i+j\leq n} f_{i+j+1}\circ(1^{\otimes i}\otimes D_{n-i-j}\otimes 1^{\otimes j}) = \sum_{i_1+\ldots+i_r=n} D_r\circ (f_{i_1}\otimes\ldots f_{i_r}).

For example, f 1D 1=D 1f 1f_1\circ D_1 = D_1\circ f_1.

Rectification

Theorem

(Kadeishvili (1980), Merkulov (1999))

If AA is a dg-algebra, regarded as a strictly associative A A_\infty-algebra, its chain cohomology H (A)H^\bullet(A), regarded as a chain complex with trivial differentials, naturally carries the structure of an A A_\infty-algebra, unique up to isomorphism, and is weakly equivalent to AA as an A A_\infty-algebra.

More details are at Kadeishvili's theorem.

Remark

This theorem provides a large supply of examples of A A_\infty-algebras: there is a natural A A_\infty-algebra structure on all cohomologies such as

etc.

In Topological space

An A A_\infty-algebra in Top is also called an A-∞ space .

Examples

Every loop space is canonically an A-∞ space. (See there for details.)

Rectification

Theorem

Every A A_\infty-space is weakly homotopy equivalent to a topological monoid.

This is a classical result by (Stasheff, BoardmanVogt). It follows also as a special case of the more general result on rectification in a model structure on algebras over an operad (see there).

In spectra

See ring spectrum and algebra spectrum.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq Γ-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object

algebraic deformation quantization

dimensionclassical field theoryLagrangian BV quantum field theoryfactorization algebra of observables
general nnP-n algebraBD-n algebra?E-n algebra
n=0n = 0Poisson 0-algebraBD-0 algebra? = BD algebraE-0 algebra? = pointed space
n=1n = 1P-1 algebra = Poisson algebraBD-1 algebra?E-1 algebra? = A-∞ algebra

References

A survey of A A_\infty-algebras in chain complexes is in

Classical articles on A A_\infty-algebra in topological spaces are

  • Jim Stasheff, Homotopy associativity of H-spaces I , Trans. Amer. Math. Soc. 108 (1963), p. 275-292.

A brief survey is in section 1.8 of

  • Martin Markl, Steve Shnider, James D. Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.

Revised on June 19, 2013 20:20:49 by Urs Schreiber (82.169.65.155)