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295
Symplectic reflection algebras, CalogeroMoser space, and deformed HarishChandra homomorphism
 Invent. Math
"... To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic ..."
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Cited by 182 (38 self)
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To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is expected to be related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/Γ. If Γ is the Weyl group of a root system in a vector space h and V = h ⊕ h ∗ , then the algebras Hκ are certain ‘rational ’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group of g = gl n. We construct a 1parameter deformation of the HarishChandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n, to the algebra of Sninvariant differential operators on the Cartan subalgebra C n with rational coefficients. The second order Laplacian on g goes, under our deformed homomorphism, to the CalogeroMoser
Categorical homotopy theory
 Homology, Homotopy Appl
"... This paper is an exposition of the ideas and methods of Cisinksi, in the context of Apresheaves on a small ..."
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Cited by 168 (7 self)
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This paper is an exposition of the ideas and methods of Cisinksi, in the context of Apresheaves on a small
Isomorphism conjectures in algebraic Ktheory
 J. Amer. Math. Soc
, 1993
"... 1.1 The Isomorphism Conjecture in algebraic Ktheory........ 2 1.2 Main Results and Corollaries.................... 4 1.3 A brief outline............................ 6 ..."
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Cited by 110 (12 self)
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1.1 The Isomorphism Conjecture in algebraic Ktheory........ 2 1.2 Main Results and Corollaries.................... 4 1.3 A brief outline............................ 6
Triangulated categories of singularities and Dbranes in LandauGinzburg models
 Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262
, 2005
"... Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend ..."
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Cited by 95 (4 self)
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Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend
Stability conditions on triangulated categories
, 212
"... This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas’s notion of Πstability. From a mathematical point of view, the most interesting feature of the definition ..."
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Cited by 80 (6 self)
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This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas’s notion of Πstability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability
Deriving Dg Categories
, 1993
"... We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapt ..."
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Cited by 75 (9 self)
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We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem` (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories. Summary We give an account of the contents of this paper for the special case of DG algebras. Let k be a commutative ring and A a DG (k)algebra, i.e. a Zgraded kalgebra A = a p2Z A p endowed with a differential d of degree 1 such that d(ab) = (da)b + (\Gamma1) p a(db) for all a 2 A p , b 2 A. A DG (right) Amodule is a Zgraded Amodule M = ` p2Z M p endowed with a differential d of degree 1 such that d(ma) = (dm)a + (\Gamma1) p m(da) for all m 2 M p , a 2 A. A morphism of DG Amodules is a homogeneous morphism of degree 0 of the underlying graded Amodules commuting with the differentials. The DG Amodules form an abelian category CA. A morphism f : M ! N of CA is nullhomotopic if f = dr + rd for some homogeneous morphism r : M ! N of degree1 of the underlying graded Amodules.
CHAIN COMPLEXES AND STABLE CATEGORIES
 MANUS. MATH.
, 1990
"... Under suitable assumptions, we extend the inclusion of an additive ... complexes concentrated in positive degrees. We thereby obtain a new proof for the key result of J. Rickard’s ’Morita theory for Derived categories ‘ [17] and a sharpening of a theorem of Happel [12, 10.10] on the ’moduletheoreti ..."
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Cited by 52 (8 self)
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Under suitable assumptions, we extend the inclusion of an additive ... complexes concentrated in positive degrees. We thereby obtain a new proof for the key result of J. Rickard’s ’Morita theory for Derived categories ‘ [17] and a sharpening of a theorem of Happel [12, 10.10] on the ’moduletheoretic description ‘ of the derived
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
"... ..."
Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Cited by 47 (0 self)
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
On the Cyclic Homology of Exact Categories
 JPAA
"... The cyclic homology of an exact category was defined by R. McCarthy [26] using the methods of F. Waldhausen [36]. McCarthy's theory enjoys a number of desirable properties, the most basic being the agreement property, i.e. the fact that when applied to the category of finitely generated projective m ..."
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Cited by 45 (1 self)
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The cyclic homology of an exact category was defined by R. McCarthy [26] using the methods of F. Waldhausen [36]. McCarthy's theory enjoys a number of desirable properties, the most basic being the agreement property, i.e. the fact that when applied to the category of finitely generated projective modules over an algebra it specializes to the cyclic homology of the algebra. However, we show that McCarthy's theory cannot be both compatible with localizations and invariant under functors inducing equivalences in the derived category. This is our motivation for introducing a new theory for which all three properties hold: extension, invariance and localization. Thanks to these properties, the new theory can be computed explicitly for a number of categories of modules and sheaves.