## Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato (1999)

Citations: | 119 - 6 self |

### BibTeX

@MISC{Kontsevich99deformationsof,

author = {Maxim Kontsevich and Yan Soibelman},

title = {Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

The deformation theory of associative algebras is a guide for developing the deformation theory of many algebraic structures. Conversely, all the

### Citations

376 | Gromov-Witten classes, quantum cohomology, and enumerative geometry - Kontsevich, Yu - 1994 |

243 |
Koszul duality for operads
- Ginzburg, Kapranov
- 1994
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Citation Context ...e thanks IHES for hospitality and stimulating research atmosphere. 1 Generalities on operads 1.1 Polynomial functors, operads, algebras The material of this subsection is very well-known (see for ex. =-=[GiKa]-=-, [GeJ], [GeKa], [Ma]). We present it here for completeness and in order to fix the 3notation. Let k be a field of characteristic zero. All vector spaces below will be k-vector spaces unless we say o... |

219 |
On quasitriangular quasi-Hopf algebras and a group closely connected with
- Drinfel'd
- 1991
(Show Context)
Citation Context ... short) and its role in the deformation theory. The Grothendieck-Teichmüller group can be defined as the automorphism group of the tower of the pro-nilpotent completions of the pure braid groups (see =-=[Dr]-=-). The pure braid group of n strings is the fundamental group of the 1configuration space of n points in the plane. Deligne’s conjecture shows its relation to the Hochschild complex. More evidence fo... |

198 |
Homotopy invariant algebraic structures on topological spaces
- Boardman, Vogt
- 1973
(Show Context)
Citation Context ...s based on the notion of a free resolution of an operad. This is the subject of Section 4. We construct canonical free resolutions via an approach similar to the one developed by Boardman and Vogt in =-=[BV]-=-. We 2use free resolutions of operads instead of the conventional approach which uses free resolutions of algebras. This approach to the deformation theory of algebras over operads is not used very o... |

181 |
A compactification of configuration space
- Fulton, MacPherson
- 1994
(Show Context)
Citation Context ...operad M which acts on the Hochschild complex of an A∞-algebra. It turns out that M is closely related to the compactifications of configuration spaces of point introduced by Fulton and Macpherson in =-=[FM]-=-. Section 6 is devoted to the proof of the theorem from Section 3. Section 7 is devoted to Deligne’s conjecture and its proof. The proof presented in this paper is based on the general deformation the... |

151 |
Quadratic Differentials
- Strebel
- 1984
(Show Context)
Citation Context ... in Section 6. 5.5 Signs in the minimal operad Now we would like to discuss the second description of the operad M. It is based on the following result from the theory of Strebel’s differentials (see =-=[St]-=-). Theorem 2 Let I = {z1, ..., zn} be a finite non-empty subset of the complex line C. Then there exists a unique quadratic differential α = f(z)(dz) 2 with f(z) = ∑ i ai z−zi , ai ∈ C, i = 1, ..., n ... |

128 |
homotopy algebra, and iterated integrals for double loop spaces
- Getzler, Jones, et al.
- 1995
(Show Context)
Citation Context ... IHES for hospitality and stimulating research atmosphere. 1 Generalities on operads 1.1 Polynomial functors, operads, algebras The material of this subsection is very well-known (see for ex. [GiKa], =-=[GeJ]-=-, [GeKa], [Ma]). We present it here for completeness and in order to fix the 3notation. Let k be a field of characteristic zero. All vector spaces below will be k-vector spaces unless we say otherwis... |

121 | Operads and motives in deformation quantization
- Kontsevich
- 1999
(Show Context)
Citation Context ...gebras to the geometry of configuration spaces of points on surfaces. One of its incarnations is Deligne’s conjecture. A brief history of the conjecture as well as its generalizations can be found in =-=[Ko3]-=-. Deligne’s conjecture is the second topic of our paper. The theme, which motivated the whole project, is the GrothendieckTeichmüller group (GT for short) and its role in the deformation theory. The G... |

44 |
Homotopy algebras are homotopy algebras
- Markl
(Show Context)
Citation Context ...s a Z-graded operad, and φ is a quasi-isomorphism. One wants to be sure that the deformation theory does not depend on the choice of the resolution. This can be achieved by assuming that (see for ex. =-=[M2]-=-) : a) F admits a filtration (as a polynomial functor) F = ∪j≥1F (j) , F (j) ⊂ F (j+1) such that dP(F (0) ) = 0 and dP(F (j) ) ⊂ FreeOP(F (j−1) ), j ≥ 1; b) φ : P → P (0) is a an epimorphism. All reso... |

29 |
Cyclic operads and cyclic homology. Geometry, topology, and physics
- Getzler, Kapranov
- 1995
(Show Context)
Citation Context ...or hospitality and stimulating research atmosphere. 1 Generalities on operads 1.1 Polynomial functors, operads, algebras The material of this subsection is very well-known (see for ex. [GiKa], [GeJ], =-=[GeKa]-=-, [Ma]). We present it here for completeness and in order to fix the 3notation. Let k be a field of characteristic zero. All vector spaces below will be k-vector spaces unless we say otherwise. We fi... |

27 | Multiplicative infinite loop space theory
- May
(Show Context)
Citation Context ...tality and stimulating research atmosphere. 1 Generalities on operads 1.1 Polynomial functors, operads, algebras The material of this subsection is very well-known (see for ex. [GiKa], [GeJ], [GeKa], =-=[Ma]-=-). We present it here for completeness and in order to fix the 3notation. Let k be a field of characteristic zero. All vector spaces below will be k-vector spaces unless we say otherwise. We fix a ca... |

22 | Cotangent cohomology of a category and deformations - Markl - 1996 |

18 | Rozansky-Witten invariants via formal geometry
- Kontsevich
(Show Context)
Citation Context ...anifold with a vector field d of degree +1 such that [d, d] = 0). We do not use any deep results concerning dgmanifolds. All necessary facts and definitions can be found in [Ko1] (note that in [Ko1], =-=[Ko2]-=- formal pointed dg-manifolds were called formal pointed Qmanifolds). We explain how to construct a formal pointed dg-manifold which controlls the deformation theory of an algebra over an operad. In tr... |

15 |
Vertex algebras
- Borcherds
- 1998
(Show Context)
Citation Context ... was introduced as a generalization of the notion of symmetric monoidal (=tensor) category. The terminology stresses the similarity of operads with tensor categories. Similar notion was introduced in =-=[B]-=- under the name multi-linear category. This notion is essentially equivalent to the notion of colored operad. We recall it below, using the name suggested in [BD]. Definition 5 A pseudo-tensor categor... |

14 |
Deformation quantization of Poisson manifolds,I
- Kontsevich
(Show Context)
Citation Context ... its relation to the Hochschild complex. More evidence for that relation has been found in deformation quantization. The coefficients in the explicit formulas for the deformed multiplication given in =-=[Ko1]-=- are periods of algebraic varieties defined over the field of rational numbers. Through the hypothetical relation to mixed Tate motives this fact leads to some conjectures about the action of the Grot... |

12 |
Chiral algebras, preprint
- Beilinson, Drinfeld
(Show Context)
Citation Context ...n of analytic functions in terms of Taylor series vs their description in terms of Taylor coefficients). 1.4 Pseudo-tensor categories The notion of colored operad has been rediscovered many times. In =-=[BD]-=- the notion of pseudo-tensor category was introduced as a generalization of the notion of symmetric monoidal (=tensor) category. The terminology stresses the similarity of operads with tensor categori... |

12 | Formality of chain operad of small squares
- Tamarkin
(Show Context)
Citation Context ...y. It seems the proof admits a generalization to the higher-dimensional version of Deligne’s conjecture proposed in [Ko3]. Other approaches to the original Deligne’s conjecture were proposed in [MS], =-=[T]-=-, [V]. The Appendix is devoted to a theory of piecewise algebraic chains. It is suitable for real semialgebraic manifolds with corners. A typical example is the compactification of the configuration s... |

8 | A solution of Deligne’s conjecture
- McClure, Smith
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Citation Context ...s study. It seems the proof admits a generalization to the higher-dimensional version of Deligne’s conjecture proposed in [Ko3]. Other approaches to the original Deligne’s conjecture were proposed in =-=[MS]-=-, [T], [V]. The Appendix is devoted to a theory of piecewise algebraic chains. It is suitable for real semialgebraic manifolds with corners. A typical example is the compactification of the configurat... |

2 |
Meromorphic tensor categories, q-alg/9709030
- Soibelman
(Show Context)
Citation Context ...articular, compositions of operations are associative with respect to morphisms of finite sets, and the unit morphisms satisfy the properties analogous to those of the identity morphisms (see [BD] or =-=[So]-=- for details). If A is a set, then a pseudo-tensor category is exactly the same as an A-colored operad in the tensor category V. 7If we take V to be the category of sets, and take I above to be 1-ele... |

2 |
Homotopy Gerstenhaber algebras, math.QA/9908040 Authors addresses: I.H.E.S., 35 route de Chartres, Bures-sur-Yvette 91440
- Voronov
(Show Context)
Citation Context ...d), one can invert the quasi-isomorphism mentioned in the theorem. Since the Hochschild complex is a P-algebra, we obtain the following Corollary known as Deligne’s conjecture (see for example [Ko3], =-=[V]-=-, [MS]). 47Corollary 1 a) The Hochschild complex C · (A, A) of an A∞-algebra A can be equipped with a structure of an algebra over the operad Chains(FM2). b) The corresponding structure of a H · (Cha... |