## A-Infinity Algebras in Representation Theory (2001)

Citations: | 6 - 0 self |

### BibTeX

@MISC{Keller01a-infinityalgebras,

author = {Bernhard Keller},

title = {A-Infinity Algebras in Representation Theory},

year = {2001}

}

### OpenURL

### Abstract

We give a brief introduction to A1-algebras and show three contexts in which they appear in representation theory: the study of Yoneda algebras and Koszulity, the description of categories of ltered modules and the description of triangulated categories. Contents 1. Denitions, the bar construction, the minimality theorem 1 2. Yoneda algebras, Koszulity and ltered modules 5 3. Description of triangulated categories 8 References 10 1. Definitions, the bar construction, the minimality theorem 1.1. A-innity algebras and morphisms. We refer to [11] for a list of references and a topological motivation for the following denition: Let k be a eld. An A1 - algebra over k is a Z-graded vector space A = M p2Z A p endowed with graded maps (=homogeneous k-linear maps) mn : A

### Citations

342 | Homological algebra of mirror symmetry
- Kontsevich
- 1994
(Show Context)
Citation Context ...n H-unital A1 -category A such that there is a bijection A ! UA from the objects of A to those of U and a triangle equivalence D per A ! T mapping the free A-module HomA (?; A) to UA . 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, !X is then trivial). Kontsevic... |

130 | Braid group actions on derived categories of coherent sheaves - Seidel, Thomas |

80 | Categorical mirror symmetry: the elliptic curve
- Polishchuk, Zaslow
- 1998
(Show Context)
Citation Context ...d be Floer homology groups and higher compositions given by Donaldson products. No rigorous denition 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 int... |

59 | Strongly homotopy algebras of a Kähler manifold
- Merkulov
(Show Context)
Citation Context ...isms of A1 -algebras f : A ! B are in bijection with morphisms of dg coalgebras TSA ! TSB. 4 BERNHARD KELLER 1.3. The minimality theorem. Theorem (Kadeishvili [6], see also [5], [21], [19], [2], [4], =-=[1-=-5]). Let A be an A1 - algebra. Then the homology H A has an A1 -algebra structure such that 1) we have m 1 = 0 and m 2 is induced by m A 2 and 2) there is a quasi-isomorphism of A1 -algebras H A ! A... |

56 | Hypergeometric functions and mirror symmetry in toric varieties
- Horja
(Show Context)
Citation Context ...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]. Rened versions of Kontsevich's conjecture avoid many of the analytic problems that plagued his earlier approach. We refer to Kontsevich-Soibelman [13] for detail... |

33 |
Excision in cyclic homology and in rational algebraic Ktheory
- Wodzicki
- 1989
(Show Context)
Citation Context ...degree 1 i satisfying certain conditions. Thesrst of these states that f 1 is a morphism of complexes. By denition, f is a quasi-isomorphism if f 1 is a quasi-isomorphism. An A1-module M is H-unital [=-=24]-=- if the corresponding comodule M k TSA has vanishing homology. If A is a unital associative algebra concentrated in degree 0 and M a unital module over A, then M TSA coincides with the augmented bar r... |

15 | Transferring algebra structures up to homology equivalence
- Johansson, Lambe
(Show Context)
Citation Context ...morphisms of A1 -algebras f : A ! B are in bijection with morphisms of dg coalgebras TSA ! TSB. 4 BERNHARD KELLER 1.3. The minimality theorem. Theorem (Kadeishvili [6], see also [5], [21], [19], [2], =-=[-=-4], [15]). Let A be an A1 - algebra. Then the homology H A has an A1 -algebra structure such that 1) we have m 1 = 0 and m 2 is induced by m A 2 and 2) there is a quasi-isomorphism of A1 -algebras H ... |

9 | Homotopy associativity of H-spaces - Stashe¤ - 1963 |

7 |
Perturbation theory in di®erential homological algebra ii
- Gugenheim, Lambe, et al.
- 1991
(Show Context)
Citation Context ...then morphisms of A1 -algebras f : A ! B are in bijection with morphisms of dg coalgebras TSA ! TSB. 4 BERNHARD KELLER 1.3. The minimality theorem. Theorem (Kadeishvili [6], see also [5], [21], [19], =-=[2-=-], [4], [15]). Let A be an A1 - algebra. Then the homology H A has an A1 -algebra structure such that 1) we have m 1 = 0 and m 2 is induced by m A 2 and 2) there is a quasi-isomorphism of A1 -algebra... |

4 |
The algebraic structure in the homology of an A(1)-algebra
- Kadeishvili
- 1982
(Show Context)
Citation Context ...t if A and B are A1 -algebras, then morphisms of A1 -algebras f : A ! B are in bijection with morphisms of dg coalgebras TSA ! TSB. 4 BERNHARD KELLER 1.3. The minimality theorem. Theorem (Kadeishvili =-=[6-=-], see also [5], [21], [19], [2], [4], [15]). Let A be an A1 - algebra. Then the homology H A has an A1 -algebra structure such that 1) we have m 1 = 0 and m 2 is induced by m A 2 and 2) there is a q... |

2 |
The category of dierential coalgebras and the category of A(1)-algebras (Russian), Trudy Tbiliss
- Kadeishvili
- 1985
(Show Context)
Citation Context ...ategory (especially if we disregard the signs). A more conceptual proof of this uses the bar construction. A-INFINITY ALGEBRAS IN REPRESENTATION THEORY 3 1.2. The bar construction. Following [23] and [7] we will restate the denitions of A1 -algebras and their morphisms in a more ecient way. Let V be a Z-graded vector space and let TV = k V V 2 be the reduced tensor algebra (in the catego... |

2 | On derived categories of representations categories - Ovsienko - 1985 |

2 | Bimodule and matrix problems - Ovsienko - 1998 |

2 |
Algebres dierentielles fortement homotopiquement associatives
- Proute
- 1984
(Show Context)
Citation Context ...bras, then morphisms of A1 -algebras f : A ! B are in bijection with morphisms of dg coalgebras TSA ! TSB. 4 BERNHARD KELLER 1.3. The minimality theorem. Theorem (Kadeishvili [6], see also [5], [21], =-=[19-=-], [2], [4], [15]). Let A be an A1 - algebra. Then the homology H A has an A1 -algebra structure such that 1) we have m 1 = 0 and m 2 is induced by m A 2 and 2) there is a quasi-isomorphism of A1 -al... |

1 |
On the theory of homology of spaces
- Kadeishvili
- 1980
(Show Context)
Citation Context ...re A1 -algebras, then morphisms of A1 -algebras f : A ! B are in bijection with morphisms of dg coalgebras TSA ! TSB. 4 BERNHARD KELLER 1.3. The minimality theorem. Theorem (Kadeishvili [6], see also =-=[5-=-], [21], [19], [2], [4], [15]). Let A be an A1 - algebra. Then the homology H A has an A1 -algebra structure such that 1) we have m 1 = 0 and m 2 is induced by m A 2 and 2) there is a quasi-isomorphi... |

1 | Twisted tensor products for the category of A(1)-algebras and A(1)- modules (Russian, English summary), Trudy Tbiliss - Kadeishvili - 1986 |

1 | The functor D for a category of A(1)-algebras - Kadeishvili - 1987 |

1 | The structure of the A(1)-algebra, and the Hochschild and Harrison cohomologies (Russian), Trudy Tbiliss - Kadeishvili - 1988 |

1 |
Introduction to A-in algebras and modules
- Keller
(Show Context)
Citation Context ...oszulity andsltered modules 5 3. Description of triangulated categories 8 References 10 1. Definitions, the bar construction, the minimality theorem 1.1. A-innity algebras and morphisms. We refer to [=-=11-=-] for a list of references and a topological motivation for the following denition: Let k be aseld. An A1 - algebra over k is a Z-graded vector space A = M p2Z A p endowed with graded maps (=homogeneo... |

1 | Homological mirror symmetry and torus available at http://xxx.lanl.gov/abs/math.SG/0011041 - Kontsevich, Soibelman |

1 |
On A1 -categories
- Lefevre
(Show Context)
Citation Context ...or 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. =-=[1-=-4]. Proposition. The category of twisted stalks twst Ext is equivalent to F by a functor that sends the object (M i ;s= 0) to M i , 1 i r. 8 BERNHARD KELLER In the case where M 1 ; : : : ; M r are t... |

1 |
Homology of spaces (Russian), Uspekhi Mat. Nauk 35
- Smirnov
- 1980
(Show Context)
Citation Context ... -algebras, then morphisms of A1 -algebras f : A ! B are in bijection with morphisms of dg coalgebras TSA ! TSB. 4 BERNHARD KELLER 1.3. The minimality theorem. Theorem (Kadeishvili [6], see also [5], =-=[21-=-], [19], [2], [4], [15]). Let A be an A1 - algebra. Then the homology H A has an A1 -algebra structure such that 1) we have m 1 = 0 and m 2 is induced by m A 2 and 2) there is a quasi-isomorphism of ... |