### Citations

876 |
Deformation quantization of Poisson manifolds
- Kontsevich
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Citation Context ...rmula is tractable! This is a good exercise!) Example 2.12. Next, let {−,−} be a linear Poisson bracket on g∗ = An, i.e., a Lie bracket on the vector space g = kn of linear functions. As explained in =-=[Kon03]-=- (see Exercise 1 of Exercise Sheet 4), if (OAn [[~]], ⋆) is Kontsevich’s canonical quantization, then x ⋆ y − y ⋆ x = [x, y], so that, as in Exercise 2.8, the map which is the identity on linear funct... |

777 | An introduction to homological algebra, Cambridge - Weibel - 1994 |

667 |
Reflection Groups and Coxeter Groups, Cambridge Stud
- Humphreys
- 1990
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Citation Context ...Corollary 4.52. 4 The degrees di of the generators gr xi are known as the fundamental degrees, and they satisfy di = mi + 1 where mi are the Coxeter exponents of the associated root system, cf. e.g., =-=[Hum90]-=-). By the theorem, for every character η : Z(Ug)→ k, one obtains an algebra (Ug)η which quantizes (Sym g)/((Sym g)g+). Here (Sym g) g + is the augmentation ideal of Sym g, which equals gr(ker η) since... |

365 | Koszul duality patterns in Representation Theory
- Beilinson, Ginzburg, et al.
- 1996
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Citation Context ...]]/(E) is a flat formal deformation of B if and only if it is flat in graded degree three, i.e., A3 ∼= B3[[~]] as a k[[~]]-module. Proof. For you to do if you know Koszul complexes (or look at, e.g., =-=[BGS96]-=-: the key is that flatness in degree ≤ 3 (or equal to 3 in the second case) is equivalent to saying that the Koszul complex of B deforms to a complex; then we apply the exercise. (6) Next, we play w... |

277 | Symplectic reflection algebras, Calogero-Moser space, and deformed HarishChandra homomorphism
- Etingof, Ginzburg
(Show Context)
Citation Context ...zed by elements c ∈ HH2(A) as above. Let S be the set of symplectic reflections, i.e., elements such taht rk(g − Id) = 2. Then c ∈ k[S]G. This deformation is the symplectic reflection algebra H1,c(G) =-=[EG02]-=-, first constructed by Drinfeld. Explicitly, this deformation is presented by TV/ ( xy − yx− ω(x, y) + 2 ∑ s∈S c(s)ωs(x, y) · s ) , 17 where ωs is the composition of ω with the projection to the sum o... |

89 | Deformation Quantization Of Algebraic Varieties
- Kontsevich
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Citation Context ...of Kontsevich’s formality theorem. Remark 4.32. Kontsevich proved this result for Rn or smooth C∞ manifolds; for the general smooth affine setting, when k contains R, one can extract this result from =-=[Kon01]-=-; for more details see [Yek05], and also, e.g., [VdB06]. These proofs also yield a sheaf-level version of the statement for the nonaffine algebraic setting. For a simpler proof in the affine algebraic... |

77 |
A relation between Hochschild homology and cohomology for Gorenstein rings
- Bergh
- 1998
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Citation Context ... Hochschild dimension d, because under this assumption, the Van den Bergh duality theorem (Theorem 5.3) still holds, and that implies that HHi(A,M) = 0 for i > d and all bimodules M . Remark 3.30. In =-=[VdB98]-=-, Van den Bergh shows that, if A has Hochschild dimension d and that A is finitely-generated and bimodule coherent, i.e., any morphism of finitely generated free Abimodules has a finitely-generated ke... |

76 | Lie theory for nilpotent L∞-algebras
- Getzler
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Citation Context ... morphisms is to study what we require to obtain a functor on Maurer-Cartan elements. We will study this generally for two arbitrary dglas, g and h. First of all, we obviously have: Proposition 4.36. =-=[Get09]-=- Any dgla morphism F : g→ h induces a functorial map in augmented pronilpotent dg commutative rings R = k⊕R+, F : MCE(g⊗̂kR)→ MCE(h⊗̂kR). 30 However, it is not true that all functorial maps are obtain... |

39 |
Homologie des invariants d’une algèbre de Weyl sous l’action d’un groupe fini
- Alev, Farinati, et al.
- 1993
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Citation Context ...ted star products yield all possible formal deformations of A up to gauge equivalence. Example 2.27. If A = Weyl(V )⋊G for G < Sp(V ) a finite subgroup, then A has finite Hochschild dimension, and by =-=[AFLS00]-=-, HHi(A) is the space of conjugation-invariant functions on the set of group elements g ∈ G such that rk(g−Id) = i. In particular, HHi(A) = 0 when i is odd, and HH2(A) is the space of conjugation-inva... |

28 | The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal
- Dolgushev, Tamarkin, et al.
- 2007
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Citation Context ...so yield a sheaf-level version of the statement for the nonaffine algebraic setting. For a simpler proof in the affine algebraic setting, which works over arbitrary fields of characteristic zero, see =-=[DTT07]-=-. That proof uses operadic machinery. The one parameter version of the theorem is Theorem 4.33. [Kon03, Kon01, Yek05, DTT07] There is a map Formal Poisson structures on X → Formal deformations of OX w... |

28 | Deformation quantization in algebraic geometry
- Yekutieli
- 2005
(Show Context)
Citation Context ...rem. Remark 4.32. Kontsevich proved this result for Rn or smooth C∞ manifolds; for the general smooth affine setting, when k contains R, one can extract this result from [Kon01]; for more details see =-=[Yek05]-=-, and also, e.g., [VdB06]. These proofs also yield a sheaf-level version of the statement for the nonaffine algebraic setting. For a simpler proof in the affine algebraic setting, which works over arb... |

27 | Poincare-BirkhoffWitt deformations of Calabi-Yau algebras
- Berger, Taillefer
- 2007
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Citation Context ... + c 3 [x3 + y3 + z3]. Example 5.16. NCCR examples? See Wemyss’s lectures! 5.5. Deformations of potentials and PBW theorems. The first part of the following theorem was proved in the filtered case in =-=[BT07]-=- and in the formal case in [EG10]. I have written informal notes proving the converse (the second part). Theorem 5.17. Let AΦ be a graded CY algebra defined by a (CY) potential Φ. Then for any filtere... |

25 | Cherednik and Hecke algebras of varieties with a finite group action
- Etingof
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Citation Context ...acts by the block matrix( A 0 0 (At)−1 ) , with At denoting the transpose of A.) 8 Preview: By deforming Diff(X)G, one obtains global versions of the spherical rational Cherednik algebras, studied in =-=[Eti04]-=-. Example 1.36. Let X = A1 \ {0} and let G = {1, g} where g(x) = x−1. Then (T ∗X)/G is a “global” or “multiplicative” version of the variety A2/(Z/2) of Example 1.27. The quantization is DGX , and one... |

23 |
Kontsevich star product on the dual of a Lie algebra
- Dito
- 1999
(Show Context)
Citation Context ...ich’s canonical quantization, then x ⋆ y − y ⋆ x = [x, y], so that, as in Exercise 2.8, the map which is the identity on linear functions yields an isomorphism U~(g)→ (Og∗ [[~]], ⋆). It turns out, by =-=[Dit99]-=-, that, up to applying a gauge equivalence to Kontsevich’s star-product (see §4.4 below), the inverse to this is again the symmetrization map of Exercise 2.8.(b). The latter star-product is called the... |

20 | On global deformation quantization in the algebraic case
- Bergh
(Show Context)
Citation Context ...ich proved this result for Rn or smooth C∞ manifolds; for the general smooth affine setting, when k contains R, one can extract this result from [Kon01]; for more details see [Yek05], and also, e.g., =-=[VdB06]-=-. These proofs also yield a sheaf-level version of the statement for the nonaffine algebraic setting. For a simpler proof in the affine algebraic setting, which works over arbitrary fields of characte... |

18 | Finite dimensional representations of symplectic reflection algebras associated to wreath products. Represent. Theory 9
- Etingof, Montarani
- 2005
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Citation Context ... space K at the origin, which is a module over (A⊗OS , ⋆|S). Moreover, if HH 1(A,EndM) = 0, then this deformation M~ is unique up to OS -linear isomorphisms which are the identity modulo (OS)+. 24 In =-=[EM05]-=-, this was used to show the existence of a unique family of irreducible representations of a wreath product Cherednik algebra H1,(k,c)(Γ n⋊Sn) for Γ < SL2(C) finite, deforming a module of the form Y ⊗... |

12 | Cyclic homology, Grundlehren der - Loday - 1998 |

6 |
produit star de Kontsevich sur le dual d’une algèbre de Lie nilpotente
- Arnal, Le
- 1998
(Show Context)
Citation Context ...and is very complicated to write out explicitly (it probably requires the Campbell-Baker-Hausdorff formula). For a description of this product, see, e.g., [Dit99, (13)]. Moreover, as first noticed in =-=[Arn98]-=- (see also [Dit99]), there is no gauge equivalence required when g is nilpotent, i.e., in this case the Kontsevich star-product equals the Gutt product (this is essentially because nothing else can ha... |

6 | Formality theorems for Hochschild complexes and their - Dolgushev, Tamarkin, et al. - 2009 |

6 |
Homology Associated with Poisson Structures, in: Deformation Theory and Symplectic Geometry, Eds. D. Sternhemer et al
- Mathieu
- 1997
(Show Context)
Citation Context ...on. Moreover, there is a canonical bijection, up to isomorphisms equal to the identity modulo ~, between deformation quantizations and formal Poisson deformations. Remark 2.10. As shown by O. Mathieu =-=[Mat97]-=-, in general there are obstructions to the existence of a quantization. We explain this following §1.4 of www.math.jussieu.fr/~keller/emalca.pdf, which more generally forms a really nice reference for... |

4 |
A Note on Br-infinity and KS-infinity formality
- Willwacher
(Show Context)
Citation Context ...he center of the quantization on the RHS. This is highly nontrivial and was proved by Kontsevich in [Kon03, §8]. More conceptually, the reason why this holds is that, as demonstrated by Willwacher in =-=[Wil11]-=-, Kontsevich’s L∞ quasi-isomorphism HH •(OX ) ∼→ C•(OX) actually can be lifted to a G∞ morphism 33 (and hence quasi-isomorphism), where G∞ is the Gerstenhaber algebra version of L∞, i.e., which takes ... |

1 | Exploring noncommutative algebras via representation theory - Etingof - 2005 |