A-INFINITY ALGEBRAS IN REPRESENTATION THEORY

BERNHARD KELLER

Abstract. We give a brief introduction to A_{∞}-algebras and show three contexts in which they appear in representation theory: the study of Yoneda algebras and Koszulity, the description of categories of filtered modules and the description of triangulated categories.

Contents

1. Definitions, the bar construction, the minimality theorem 1 2. Yoneda algebras, Koszulity and filtered modules 5 3. Description of triangulated categories 8 References 10

1. Definitions, the bar construction, the minimality theorem

1.1. A-infinity algebras and morphisms. We refer to [11] for a list of references and a topological motivation for the following definition: Let k be a field. An A_{∞}- algebra over k is a Z-graded vector space

A = ⨁_{p∈Z} A^{p}

endowed with graded maps (=homogeneous k-linear maps)

m_{n} : A^{⊗n} → A , n ≥ 1 ,

of degree 2 − n satisfying the following relations • We have m_{1} m_{1} = 0, i.e. (A, m_{1}) is a differential complex.

• We have

m_{1} m_{2} = m_{2} (m_{1} ⊗ 1 + 1 ⊗ m_{1})

as maps A^{⊗2} → A. Here 1 denotes the identity map of the space A. So m_{1} is a (graded) derivation with respect to the multiplication m_{2}.

• We have

m_{2}(1 ⊗ m_{2} − m_{2} ⊗ 1)

= m_{1}m_{3} + m_{3} (m_{1} ⊗ 1 ⊗ 1 + 1 ⊗ m_{1} ⊗ 1 + 1 ⊗ 1 ⊗ m_{1}) as maps A^{⊗3} → A. Note that the left hand side is the associator for m_{2} and that the right hand side may be viewed as the boundary of m_{3} in the morphism complex Hom^{•}_{k} (A^{⊗3}, A). This implies that m_{2} is associative up to homotopy.

Date: March 24, 2001, last modified on March 24, 2001.

1991 Mathematics Subject Classification. 18E30, 16D90, 18G40, 18G10, 55U35.

Key words and phrases. A-infinity algebra, Derived category.

Partially supported by the European Network ‘Algebraic Lie Representations’, Contract ERB-FMRX-CT97-0100.

• More generally, for n ≥ 1, we have

∑(−1)^{r+st} m_{u} (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t}) = 0

where the sum runs over all decompositions n = r + s + t and we put u = r + 1 + t.

More signs appear when these formulas are applied to elements by the Koszul sign rule

(f ⊗ g)(x ⊗ y) = (−1)^{|g||x|}f(x) ⊗ g(y)

for homogeneous maps f, g and homogeneous elements x, y, where the vertical bars denote the degree. In general, an A_{∞}-algebra is not associative, but its homology H^{∗}A is a (graded) associative algebra for the multiplication induced by m_{2}. If A = A^{0}, i.e. A^{p} = 0 for all p ≠ 0, then A is an ordinary associative algebra (but not necessarily unital) and conversely, each graded associative algebra may be viewed as an A_{∞}-algebra whose compositions m_{i} vanish for i ≥ 3.

A morphism of A_{∞}-algebras f : A → B is a family

f_{n} : A^{⊗n} → B

of graded maps of degree 1 − n such that

• we have f_{1}m_{1} = m_{1}f_{1}, i.e. f_{1} is a morphism of complexes,

• we have

f_{1}m_{2} = m_{2} (f_{1} ⊗ f_{1}) + m_{1}f_{2} + f_{2} (m_{1} ⊗ 1 + 1 ⊗ m_{1}) ,

which means that f_{1} commutes with the multiplication m_{2} up to a homotopy given by f_{2},

• more generally, for n ≥ 1, we have

∑(−1)^{r+st} f_{u} (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t}) = ∑(−1)^{s} m_{r} (f_{i_{1}} ⊗ f_{i_{2}} ⊗ … ⊗ f_{i_{r}})

where the first sum runs over all decompositions n = r + s + t, we put u = r+1+t, and the second sum runs over all 1 ≤ r ≤ n and all decompositions n = i_{1} + ⋯ + i_{r}; the sign on the right hand side is given by

s = (r − 1)(i_{1} − 1) + (r − 2)(i_{2} − 1) + ⋯ + 2(i_{r−2} − 1) + (i_{r−1} − 1).

The morphism f is a quasi-isomorphism if f_{1} is a quasi-isomorphism. It is strict if f_{i} = 0 for all i ≠ 1. The identity morphism is the strict morphism f such that f_{1} is the identity of A. The composition of two morphisms f : B → C and g : A → B is given by

(f ∘ g)_{n} = ∑(−1)^{s}f_{r} ∘ (g_{i_{1}} ⊗ ⋯ ⊗ g_{i_{r}})

where the sum and the sign are as in the defining identity. It is not hard to check we do obtain a category (especially if we disregard the signs). A more conceptual proof of this uses the bar construction.

1.2. The bar construction. Following [23] and [7] we will restate the definitions of A_{∞}-algebras and their morphisms in a more efficient way. Let V be a Z-graded vector space and let

\overline{T} V = k ⊕ V ⊕ V^{⊗2} ⊕ ⋯

be the reduced tensor algebra (in the category of graded spaces). We make \overline{T} V into a graded coalgebra via the comultiplication

Δ : \overline{T} V → \overline{T} V ⊗ \overline{T} V

defined by

Δ(v_{1}, …, v_{n}) = ∑_{1≤i≤n}(v_{1}, …, v_{i}) ⊗ (v_{i+1}, …, v_{n}) .

Note that Δ(v) = 0 and Δ(v_{1}, v_{2}) = v_{1} ⊗ v_{2}. It is then easy to see that each graded map b : \overline{T} V → V of degree 1 uniquely lifts to a graded coderivation b : \overline{T} V → \overline{T} V of degree 1, where coderivation means that we have

Δb = (b ⊗ 1 + 1 ⊗ b) Δ.

To see that this is plausible, one should remember that algebra derivations \overline{T} V → \overline{T} V are in bijection with maps V → \overline{T} V. The component of b that maps V^{⊗n} to V^{⊗u} is

∑ 1^{⊗r} ⊗ b_{s} ⊗ 1^{⊗t}

where r+s+t = n and r+1+t = u. Thus we have a bijection between coderivations b : \overline{T} V → \overline{T} V of degree 1 and families of maps b_{n} : V^{⊗n} → V, n ≥ 1, of degree 1.

Now let A be another Z-graded space and let SA be its suspension: (SA)^{p} = A^{p+1}. Define a bijection between the families of maps b_{n} : (SA)^{⊗n} → SA of degree 1 and maps m_{n} : A^{⊗n} → A, n ≥ 1, by the commutative square

\begin{matrix}(SA)^{⊗n} & →^{b_{n}} & SA \\ s^{⊗n} ↑ & & ↑ s \\ A^{⊗n} & →^{m_{n}} & A \end{matrix} ,

where s : A → SA is the canonical map a ↦ a of degree −1. Note that m_{n} is of degree −n + 1 − (−1) = 2 − n. Then we have the following easy

Lemma. The following are equivalent

(i) The maps m_{n} : A^{⊗n} → A yield an A_{∞}-algebra structure on A.

(ii) The coderivation b : \overline{T} SA → \overline{T} SA satisfies b^{2} = 0.

(iii) For each n ≥ 1, we have

∑ b_{u} (1^{⊗r} ⊗ b_{s} ⊗ 1^{⊗t}) = 0 ,

where the sum runs over all decompositions n = r + s + t and we put u = r + 1 + t.

Note that no signs appear in the last formula. The lemma yields a bijection between A_{∞}-algebra structures on A and coalgebra differentials on \overline{T} SA. It is also easy to check that if A and B are A_{∞}-algebras, then morphisms of A_{∞}-algebras f : A → B are in bijection with morphisms of dg coalgebras \overline{T} SA → \overline{T} SB.

1.3. The minimality theorem.

Theorem (Kadeishvili [6], see also [5], [21], [19], [2], [4], [15]). Let A be an A_{∞}- algebra. Then the homology H^{∗}A has an A_{∞}-algebra structure such that

1) we have m_{1} = 0 and m_{2} is induced by m_{2}^{A} and

2) there is a quasi-isomorphism of A_{∞}-algebras H^{∗}A → A lifting the identity

of H^{∗}A. Moreover, this structure is unique up to (non unique) isomorphism of A_{∞}-algebras.

Let us introduce some terminology: An A_{∞}-algebra with vanishing m_{1} is called minimal. In the situation of the theorem, the A_{∞}-algebra H^{∗}A is called a minimal model for A. The A_{∞}-algebra A is called formal if its minimal model can be chosen to be an ordinary associative graded algebra, i.e. such that m_{n} = 0 for all n ≥ 3.

As a consequence, we obtain that if B is an ordinary associative algebra with 1 and M a B-module, then Ext_{B}^{∗}(M, M) carries an A_{∞}-algebra structure canonical up to non unique A_{∞}-isomorphism. Indeed, choose a projective resolution P → M of M. Then the morphism complex A = Hom^{•}_{B}(P, P) is a differential graded algebra whose homology identifies with the Yoneda algebra Ext_{B}^{∗}(M, M). In particular, A is an A_{∞}-algebra and its homology Ext_{B}^{∗}(M, M) carries the A_{∞}-structure of the minimal model of A.

1.4. Inductive construction of the higher compositions. Let us suppose that A is a dg algebra. We will show how to construct the first four higher compositions m_{i} on H^{∗}A and the first four components f_{i} of an A_{∞}-quasi-isomorphism f : H^{∗}A → A. We will denote all composition maps with i arguments by m_{i}. In particular, m_{1} : A → A is the differential of A and m_{2} : A ⊗ A → A the multiplication, and all higher m_{i} : A^{⊗i} → A, i ≥ 3, vanish. We wish to construct the m_{i} : (H^{∗}A)^{⊗i} → H^{∗}A.

1) Let m_{1} : H^{∗}A → H^{∗}A be zero. Choose f_{1} as a morphism of complexes (H^{∗}A, 0) → A such that H^{∗}f_{1} identifies with the identity of H^{∗}A.

2) Let m_{2} : H^{∗}A ⊗ H^{∗}A → H^{∗}A be the multiplication induced by that of A. Then by definition, the morphisms m_{2} (f_{1} ⊗ f_{1}) and f_{1} m_{2} are homotopic as morphisms of complexes from H^{∗}A ⊗ H^{∗}A to A. We choose f_{2} : H^{∗}A ⊗ H^{∗}A → A as a morphism of complexes of degree −1 such that

f_{1}m_{2} = m_{1}f_{2} + m_{2} (f_{1} ⊗ f_{1}).

3) Let

Φ_{3} = m_{2} (f_{1} ⊗ f_{2} − f_{2} ⊗ f_{1}) + f_{2} (1 ⊗ m_{2} − m_{2} ⊗ 1).

Then Φ_{3} is a morphism of complexes (H^{∗}A)^{⊗3} → A of degree −1. Since f_{1} : H^{∗}A → A is a quasi-isomorphism, there is a morphism of complexes m_{3} : (H^{∗}A)^{⊗3} → H^{∗}A such that f_{1} m_{3} is homotopic to Φ_{3}. So there is a graded morphism f_{3} : (H^{∗}A)^{⊗3} → A such that

f_{1} m_{3} = m_{1} f_{3} + Φ_{3}.

4) Let

Φ_{4} = m_{2} (f_{1} ⊗ f_{3} − f_{2} ⊗ f_{2} + f_{3} ⊗ f_{1}) + f_{2} (1 ⊗ m_{3} + m_{3} ⊗ 1)
−f_{3} (1 ⊗ 1 ⊗ m_{2} − 1 ⊗ m_{2} ⊗ 1 + m_{2} ⊗ 1 ⊗ 1).

Then Φ_{4} : (H^{∗}A)^{⊗4} → A is a morphism of complexes of degree −2. Thus there is a morphism of complexes m_{4} : (H^{∗}A)^{⊗4} → H^{∗}A such that f_{1} m_{4} is homotopic to Φ_{4}. Hence we can find a graded morphism f_{4} : (H^{∗}A)^{⊗4} → A of degree −3 such that

f_{1}m_{4} = m_{1}f_{4} + Φ_{4}.

2. Yoneda algebras, Koszulity and filtered modules

2.1. Motivation. Let B be an associative algebra with 1 and let M_{1}, …, M_{r} be (right) B-modules. Consider the full subcategory F = filt(M_{1}, …, M_{r}) of the category Mod B of B-modules whose objects are the modules admitting a finite filtration whose subquotients are among the M_{i}. Note that morphisms in this category are not required to respect the filtrations. Let D Mod B denote the (unbounded) derived category of the category of B-modules and T = tria(M_{1}, …, M_{r}) its full triangulated subcategory generated by M_{1}, …, M_{r}. Put M = ⨁ M_{i}.

Question: Can we reconstruct F and T from the Yoneda algebra

Ext = Ext_{B}^{∗}(M, M) ?

The answer is clearly no. For example, if we take B = k[x]/(x^{n}), and M = k, then the graded algebra Ext identifies with the algebra k[u, v]/(v^{2}), |u| = 2, |v| = 1, for all n ≥ 3, but the categories F are non-equivalent for different values of n. However, the Yoneda algebra admits an A_{∞}-algebra structure and this structure allows us to reconstruct both F and T . More precisely,

1) there is a description of F using only

Ext^{0} ⊕ Ext^{1} ⊕ Ext^{2}

and the ‘restriction’ of the m_{i}, i ≥ 2, to this truncation. This description generalizes the description of finite-length modules over a finite-dimensional algebra as representations of a quiver with relations.

2) There is an analogous description of T which uses all of the information

present in the A_{∞}-algebra structure on Ext. Below we will give the description of F. We refer to [11] for an analogous description of T . A different description will be given below in section 3.

2.2. The Yoneda algebra of the simples. Let k be a field, Q a finite quiver and I an admissible ideal of kQ. Put B = kQ/I and let M = S be the sum of the canonical representatives of the isoclasses of the simple B-modules.

Proposition 1. a) The A_{∞}-algebra A = Ext_{B}^{∗}(S, S) is formal (i.e. A_{∞}- isomorphic to an algebra with m_{i} = 0 for all i ≥ 3) if and only if B is a Koszul algebra.

b) As an A_{∞}-algebra, Ext_{B}^{∗}(S, S) is generated by its components in degree 0 and 1.

In a), we say that B is Koszul if B is isomorphic to kQ/I′, where I′ is a homogeneous ideal (all arrows of Q are of degree 1) such that the graded algebra kQ/I′ is Koszul (cf. [1]). In b), we claim that Ext_{B}^{∗}(S, S) coincides with its smallest subspace closed under all operations m_{i} and containing the components of degree 0 and 1.

We now describe the part of the A_{∞}-structure on Ext_{B}^{∗}(S, S) which only involves the components of degree 0 and 1. For this, let J = ⟨kQ_{1}⟩ be the ideal generated by the arrows of Q. Let D denote the dual over k. Recall that we have a canonical map

⨁_{i=1}^{∞} Ext_{B}^{1}(S, S)^{⊗i} → DJ

and a canonical isomorphism

Ext_{B}^{2}(S, S) →^{∼} D(I/JI + IJ).

Let s be a k-linear section of the surjection I → I/(JI + IJ). By composing the inclusion I ⊂ J with s, we obtain a map I/(JI + IJ) → J whose image is a space of representatives for the irreducible relations. The top horizontal arrow of the following square is the dual of this map

DJ  → D(I/IJ + JI)
↑  ↑
⨁_{i=1}^{∞} Ext_{B}^{1}(S, S)^{⊗i} →^{μ} Ext_{B}^{2}(S, S)

The map μ is defined by the commutativity of this square.

Proposition 2. The component

μ_{i} : Ext_{B}^{1}(S, S)^{⊗i} → Ext_{B}^{2}(S, S)

equals the restriction of m_{i} to Ext_{B}^{1}(S, S)^{⊗i} for some choice of minimal model structure m_{i}, i ≥ 1, on Ext_{B}^{∗}(S, S).

In particular, we see that the ’restriction‘ to

Ext_{B}^{0}(S, S) ⊕ Ext_{B}^{1}(S, S) ⊕ Ext_{B}^{2}(S, S)

of the A_{∞}-structure on Ext_{B}^{∗}(S, S) yields a quiver with relations for B and hence determines the category of finite-dimensional B-modules.

2.3. Reconstruction. As announced in 2.1, we will now show how to reconstruct the category F from the A_{∞}-algebra Ext_{B}^{∗}(M, M). More precisely, we will reconstruct it from the more natural A_{∞}-category associated with the M_{i}. An A_{∞}- category A is the datum of

• a class of objects obj(A),

• for all A, B ∈ A (we write A ∈ A instead of A ∈ obj(A)), a Z-graded vector space Hom_{A}(A, B), often denoted by (A, B),

• for all n ≥ 1 and all A_{0}, …, A_{n} ∈ A, a graded map

m_{n} : (A_{n−1}, A_{n}) ⊗ (A_{n−2}, A_{n−1}) ⊗ ⋯ ⊗ (A_{0}, A_{1}) → (A_{0}, A_{n})

of degree 2 − n such that for each n ≥ 1 and all A_{0}, …, A_{n} ∈ A, we have the identity

∑(−1)^{r+st} m_{u} (1^{⊗r} ⊗ m_{s} ⊗ 1^{⊗t}) = 0

of maps

(A_{n−1}, A_{n}) ⊗ (A_{n−2}, A_{n−1}) ⊗ ⋯ ⊗ (A_{0}, A_{1}) → (A_{0}, A_{n}) ,

where the sum runs over all decompositions n = r + s + t and we put u = r + 1 + t.

For example, the datum of an A_{∞}-category A with one object ∗ is equivalent to the datum of the endomorphism A_{∞}-algebra Hom_{A} (∗, ∗). Another class of examples is provided by dg ’categories without identities‘: these are exactly the A_{∞}-categories whose composition maps m_{n} vanish for all n ≥ 3.

Note that for two reasons, an A_{∞}-category is not a category in general : firstly, the composition m_{2} may not be associative; secondly, there may not be identity morphisms.

A variant of the minimality theorem 1.3 yields a canonical (up to non unique isomorphism) structure of minimal A_{∞}-category on the graded category Ext with objects M_{1}, …, M_{r} and morphism spaces

Ext (M_{i}, M_{j}) = Ext_{B}^{∗}(M_{i}, M_{j}).

We will reconstruct F = filt(M_{1}, …, M_{r}) from Ext. In fact, as it will turn out, we only need the truncations

Ext_{B}^{0}(M_{i}, M_{j}) ⊕ Ext_{B}^{1}(M_{i}, M_{j}) ⊕ Ext_{B}^{2}(M_{i}, M_{j})

of the morphism spaces and only the restriction of the A_{∞}-structure to these truncations.

Let A be a minimal A_{∞}-category with A (A, B)^{p} = 0 for all p < 0 and all A, B ∈ A. We will construct a ‘category without units’ twst A called the category of twisted stalks. It will turn out that twst Ext is equivalent to F. A twisted stalk over A is a pair M consisting of a sequence A_{1}, …, A_{n} of objects of A and of a matrix δ = (δ_{ij}) of elements

δ_{ij} ∈ A(A_{i}, A_{j})^{1}

satisfying δ_{ij} = 0 for all i ≥ j and

0 = ∑_{t=2}^{∞}(−1)^{\frac{t(t−1)}{2}} m_{t}(δ, …, δ) = −m_{2}(δ, δ) − m_{3}(δ, δ, δ) + ….

Here we still denote by m_{i} the natural extension of m_{i} to matrices with coefficients in the morphisms spaces of A. One should think of M as an object with a filtration whose subquotients are the A_{i}.

A morphism of twisted stalks M → M′ is given by a matrix f = (f_{ij}) of elements

f_{ij} ∈ A (A_{j}, A′_{i})^{0}

satisfying

0 = −m_{2}(δ′, f) + m_{2}(f, δ) − m_{3}(δ′, δ′, f) + m_{3}(δ′, f, δ) − m_{3}(f, δ, δ) + …
±m_{n}(δ′, …, δ′, f, δ, …, δ) …

The composition of two morphisms f : M′ → M′′ and g : M → M′ is given by

m_{2}(f, g) + m_{3}(f, g, δ) − m_{3}(f, δ′, g) + m_{3}(δ′′, f, g) + …
±m_{n}(δ′′, …, δ′′, f, δ′, …, δ′, g, δ, …, δ) + …

(for the signs cf. [11, 7.6]).

Lemma. The composition of twst A is associative.

The lemma is easy to check modulo the signs. To prove it rigorously, one can use an analogue of the bar construction, cf. [14].

Proposition. The category of twisted stalks twst Ext is equivalent to F by a functor that sends the object (M_{i}, δ = 0) to M_{i}, 1 ≤ i ≤ r.

In the case where M_{1}, …, M_{r} are the simple modules over a finite-dimensional algebra B as in 2.2, this proposition reduces to the description of the category of finite-length modules over B as representations of a quiver with relations. An example with some non simple M_{i} is given in [11, 7.9].

3. Description of triangulated categories

We will explain how each ‘algebraic’ triangulated category with a generator becomes equivalent to the derived category of an A_{∞}-category. A class of conjectural examples is provided by Kontsevich’s homological interpretation of mirror symmetry.

Let A be an A_{∞}-algebra. An A_{∞}-module over A is a Z-graded space M endowed with maps

m_{n}^{M} : M ⊗ A^{⊗n} → M , n ≥ 1 ,

homogeneous of degree 2 − n which satisfy the ‘same’ identities as those in the definition of an A_{∞}-algebra (some of the m_{i} have to be interpreted as m_{i}^{M}). For example, if A is concentrated in degree 0, each complex of A^{0}-modules gives rise to an A_{∞}-module over A (but not all A_{∞}-modules are obtained in this way).

For an arbitrary A_{∞}-algebra, the structures of A_{∞}-module on a graded vector space M are in bijection with the comodule differentials on the induced comodule M ⊗_{k} T SA, where T SA = k ⊕ \overline{T} SA is the cofree coaugmented coalgebra on SA endowed with the differential of the sum of the complexes k and \overline{T} SA. By definition, the morphisms of A_{∞}-modules M → N are in bijection with the morphisms of dg comodules M ⊗T SA → N ⊗T SA. Concretely, a morphism of A_{∞}-modules M → N is given by a family of graded maps f_{i} : M ⊗A^{⊗i} → N, i ≥ 1, homogeneous of degree 1 − i satisfying certain conditions. The first of these states that f_{1} is a morphism of complexes. By definition, f is a quasi-isomorphism if f_{1} is a quasi-isomorphism.

An A_{∞}-module M is H-unital [24] if the corresponding comodule M ⊗_{k} T SA has vanishing homology. If A is a unital associative algebra concentrated in degree 0 and M a unital module over A, then M ⊗ T SA coincides with the augmented bar resolution of M over A and hence is acyclic. So M is H-unital in this case.

An A_{∞}-algebra is H-unital if it is H-unital as a module over itself. In contrast to more naive notions of unitality, H-unitality is preserved under quasi-isomorphisms. In particular, A is H-unital iff its minimal model is H-unital.

From now on, we suppose that A is an H-unital A_{∞}-algebra. The derived category D_{∞}A has as objects the H-unital A_{∞}-modules. Its morphisms are obtained from morphisms of A_{∞}-modules by formally inverting all quasi-isomorphisms. One can show [14] that D_{∞}A has a natural triangulated structure. Moreover if A is an associative unital algebra (or dg algebra) and DA its derived category, the canonical functor

DA → D_{∞}A

is an equivalence.

The perfect derived category D_{per}A is by definition the smallest triangulated subcategory of DA containing the free module A and stable under forming direct factors. Thus if A is an ordinary unital associative algebra, then D_{per}A is the subcategory of perfect complexes in DA. So in this case, it identifies with the homotopy category of bounded complexes of finitely generated projective modules. In the general case, an analogous description of D_{per}A can be given using twisted objects in a way similar to twisted stalks (2.3), cf. [11]. In fact, D_{per}A is equivalent to the closure under direct factors of the 0-homology category of the A_{∞}-category of twisted objects over A.

An algebraic triangulated category is a triangulated category T which is triangle equivalent to the stable category of a Frobenius category.

Theorem. Let T be a k-linear algebraic triangulated category with split idempotents. Suppose that T is generated by an object U, i.e. T coincides with its smallest full triangulated subcategory containing U and closed under forming direct summands. Then the graded algebra

A = ⨁_{i∈Z} Hom_{T} (U, U[i])

admits a structure of H-unital A_{∞}-algebra with m_{1} = 0 and m_{2} the natural multiplication such that there is a triangle equivalence

D_{per}A → T

mapping A to U.

For example, if T is the bounded derived category of the category of coherent sheaves on a smooth projective variety X and U is a tilting sheaf (or a tilting complex) on X, then the algebra A is concentrated in degree 0 and all the higher m_{i} vanish for degree reasons. In this case, A is a finite-dimensional algebra of finite global dimension, D_{per}A coincides with the bounded derived category of finitely generated A-modules and we obtain an equivalence

D^{b}A → D^{b}(coh X).

If B is a finite-dimensional algebra of the form kQ/I, T the bounded derived category of finite-dimensional B-modules and U the sum of the simple modules, then the A_{∞}-algebra A of the theorem identifies with the Yoneda algebra Ext_{B}^{∗}(S, S), S = U, with its (almost) canonical A_{∞}-structure. We obtain a triangle equivalence

D_{per}A → D^{b}B.

The right hand side may be thought of as a description of the derived category in terms of projective modules, whereas the left hand side is a description in terms of simple modules.

The theorem above has an obvious variant where A_{∞}-algebras are replaced by A_{∞}-categories:

Theorem. Let T be a k-linear algebraic triangulated category with split idempotents and U ⊂ T a full subcategory such that T coincides with its smallest triangulated subcategory containing U and stable under forming direct factors. Then there is an H-unital A_{∞}-category A such that there is a bijection A → U_{A} from the objects of A to those of U and a triangle equivalence

D_{per}A → T

mapping the free A-module Hom_{A}(?, A) to U_{A}.

In his lecture [12] at the ICM in 1994, M. Kontsevich gave a conjectural class of examples: Let X be a smooth projective complex variety admitting a mirror symmetric dual Y (in particular, ω_{X} is then trivial). Kontsevich conjectured that there is a triangle equivalence

D_{per}F(Y) → D^{b}(coh X)

where F(Y) is the ‘Fukaya category’ of Y. The variety Y is a symplectic variety, the objects of F(Y) are Lagrangian subvarieties L ⊂ Y endowed with additional structure, morphism spaces of F(Y) should be Floer homology groups and higher compositions given by Donaldson products. No rigorous definition seems to be known until now.

However, Kontsevich’s conjecture was proved by Polishchuk and Zaslov [18] for a suitably adapted version of F(Y) in the case where X is an elliptic curve (and so Y is a symplectic torus).

Kontsevich’s conjecture implies that the group of symplectomorphisms of Y embeds into the automorphism group of the derived category D^{b}(coh X). The existence of many automorphisms belonging conjecturally to the image of this embedding was checked in work by R. P. Horja [3] and by P. Seidel and R. Thomas [20].

Refined versions of Kontsevich’s conjecture avoid many of the analytic problems that plagued his earlier approach. We refer to Kontsevich-Soibelman [13] for details.

References

[1] A. Beilinson, V. Ginzburg and W. Soergel, J. Amer. Math. Soc. 9 (1996), 473–527.

[2] V. K. A. M. Gugenheim, L. A. Lambe, J. D. Stasheff, Perturbation theory in differential homological algebra II, Illinois J. Math. 35 (1991), 357–373.

[3] R. P. Horja, Hypergeometric functions and mirror symmetry in toric varieties, available at

http://xxx.lanl.gov/abs/math.AG/9912109.

[4] L. Johansson, L. Lambe, Transferring Algebra Structures Up to Homology Equivalence, Preprint, 1996, to appear in Math. Scand.

[5] T. V. Kadeishvili, On the theory of homology of fiber spaces (Russian), Uspekhi Mat. Nauk 35 (1980), 183–188. Translated in Russ. Math. Surv. 35 (1980), 231–238.

[6] T. V. Kadeishvili, The algebraic structure in the homology of an A(∞)-algebra (Russian), Soobshch. Akad. Nauk Gruzin. SSR 108 (1982), 249–252.

[7] T. V. Kadeishvili, The category of differential coalgebras and the category of A(∞)-algebras (Russian), Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 77 (1985), 50–70.

[8] T. V. Kadeishvili, Twisted tensor products for the category of A(∞)-algebras and A(∞)- modules (Russian, English summary), Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 83 (1986), 26–45.

[9] T. V. Kadeishvili, The functor D for a category of A(∞)-algebras (Russian), Soobshch. Akad. Nauk Gruzin. SSR 125 (1987), 273–276.

[10] T. V. Kadeishvili, The structure of the A(∞)-algebra, and the Hochschild and Harrison cohomologies (Russian), Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), 19–27.

[11] B. Keller, Introduction to A-infinity algebras and modules, Homotopy, Homology and Appli-

cations 3 (2001), 1–35, available at http://www.math.rutgers.edu/hha/

[12] M. Kontsevich, Homological algebra of Mirror Symmetry, Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994, Vol. 1, Birkhäuser Basel, 1995, 120– 139.

[13] M. Kontsevich, Y. Soibelman, Homological mirror symmetry and torus fibrations, available

at http://xxx.lanl.gov/abs/math.SG/0011041.

[14] K. Lefèvre, On A_{∞}-categories, in preparation.

[15] S. A. Merkulov, Strong homotopy algebras of a Kähler manifold, available at

http://xxx.lanl.gov/abs/math.AG/9809172.

[16] S. Ovsienko, On derived categories of representations categories, in: “XVIII Allunion algebraic conference”, Proceedings, Part II, Kishinev, (1985), 71.

[17] S. Ovsienko, Bimodule and matrix problems, Preprint, 1998.

[18] A. Polishchuk, E. Zaslow, Categorical Mirror Symmetry: The Elliptic Curve, Adv. Theor. Math. Phys. 2 (1998), no. 2, 443–470, cf. also http://xxx.lanl.gov/abs/math/9801119.

[19] A. Prouté, Algèbres différentielles fortement homotopiquement associatives, Thèse d’Etat, Université Paris VII, 1984.

[20] P. Seidel, R. Thomas, Braid group actions on derived categories of coherent sheaves, to

    appear in Duke Math. J., http://xxx.lanl.gov/abs/math.AG/0001043

[21] V. A. Smirnov, Homology of fiber spaces (Russian), Uspekhi Mat. Nauk 35 (1980), 227–230. Translated in Russ. Math. Surveys 35 (1980), 294–298.

[22] J. D. Stasheff, Homotopy associativity of H-spaces, I, Trans. Amer. Math. Soc. 108 (1963), 275–292.

[23] J. D. Stasheff, Homotopy associativity of H-spaces, II, Trans. Amer. Math. Soc. 108 (1963), 293–312.

[24] M. Wodzicki, Excision in cyclic homology and in rational algebraic K-theory, Ann. of Math. 129 (1989), 591–639.

UFR de Mathématiques, UMR 7586 du CNRS, Case 7012, Université Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France

   E-mail address: keller@math.jussieu.fr www.math.jussieu.fr/  keller