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Rational Cherednik algebras and Hilbert schemes
"... Abstract. Let Hc be the rational Cherednik algebra of type An−1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators with associated graded ring gr Uc = C[h ⊕ h ∗ ] W, where W is the nth symmetric group. Using the Zalgebra construction from [GS] it is also po ..."
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Cited by 58 (6 self)
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Abstract. Let Hc be the rational Cherednik algebra of type An−1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators with associated graded ring gr Uc = C[h ⊕ h ∗ ] W, where W is the nth symmetric group. Using the Zalgebra construction from [GS] it is also possible to associate to a filtered Hc or Ucmodule M a coherent sheaf Φ(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of Uc and Hc, and relate it to Hilb(n) and to the resolution of singularities τ: Hilb(n) → h ⊕ h ∗ /W. For example, we prove: • If c = 1/n, so that Lc(triv) is the unique onedimensional simple Hcmodule, then Φ(eLc(triv)) ∼ = OZn, where Zn = τ −1 (0) is the punctual Hilbert scheme. • If c = 1/n + k for k ∈ N then, under a canonical filtration on the finite dimensional module Lc(triv), gr eLc(triv) has a natural bigraded structure which coincides with that on H 0 (Zn, L k), where L ∼ = O Hilb(n)(1); this confirms conjectures of Berest, Etingof and Ginzburg. • Under mild restrictions on c, the characteristic cycle of Φ(e∆c(µ)) equals ∑ λ Kµλ[Zλ], where Kµλ are Kostka numbers and the Zλ are (known) irreducible components of τ −1 (h/W). Contents
Classification of Holomorphic Vector Bundles On Noncommutative Twotori
 DOCUMENTA MATH.
, 2004
"... We prove that every holomorphic vector bundle on a noncommutative twotorus T can be obtained by successive extensions from standard holomorphic bundles considered in [2]. This implies ..."
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Cited by 49 (3 self)
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We prove that every holomorphic vector bundle on a noncommutative twotorus T can be obtained by successive extensions from standard holomorphic bundles considered in [2]. This implies
A family of elliptic algebras
 Internat. Math. Res. Notices
, 1997
"... The survey is devoted to associative Z≥0graded algebras presented by n generators and n(n−1) 2 quadratic relations and satisfying the socalled PoincareBirkhoffWitt condition (PBWalgebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve ..."
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Cited by 43 (6 self)
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The survey is devoted to associative Z≥0graded algebras presented by n generators and n(n−1) 2 quadratic relations and satisfying the socalled PoincareBirkhoffWitt condition (PBWalgebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve and a point on this curve) which are flat deformations of the polynomial ring in n variables. Diverse properties of these algebras are described, together with their relations to integrable systems, deformation quantization, moduli spaces and other directions of modern investigations.
Noncommutative twotori with real multiplication as noncommutative projective varieties
 J. Geom. Phys
"... Abstract. We define analogues of homogeneous coordinate algebras for noncommutative twotori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate a ..."
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Cited by 26 (4 self)
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Abstract. We define analogues of homogeneous coordinate algebras for noncommutative twotori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative twotorus with real multiplication. These results are based on the techniques of [14] allowing to interpret all the data in terms of autoequivalences of the derived categories of coherent sheaves on elliptic curves.
Ideal classes of the Weyl algebra and noncommutative projective geometry
, 2001
"... Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le B ..."
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Cited by 26 (3 self)
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Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map θ: R → C by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that θ is inverse to a bijection ω: C → R constructed in [BW] by a completely different method. The main step in the proof is to show that θ is equivariant with respect to natural actions of the group G = Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.
Sklyanin algebras and Hilbert schemes of points
"... We construct projective moduli spaces for torsionfree sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective ..."
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Cited by 20 (2 self)
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We construct projective moduli spaces for torsionfree sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P 2. The generic noncommutative plane corresponds to the Sklyanin algebra S = Skl(E, σ) constructed from an automorphism σ of infinite order on an elliptic curve E ⊂ P 2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1 − n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P² \ E.
Quantizations of conical symplectic resolutions I: local and global structure
"... Abstract. We reexamine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical con ..."
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Cited by 19 (9 self)
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Abstract. We reexamine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the BeilinsonBernstein localization theorem, a theory of HarishChandra bimodules and their relationship to convolution operators on cohomology, and a
Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces
 Adv. Math
"... Double affine Hecke algebras for reduced root systems were introduced by Cherednik [Ch] in order to prove Macdonald conjectures. Double affine Hecke algebras of type C∨Cn were introduced in the works of Noumi, Sahi, and Stokman ([NoSt, Sa, St]) as a generalization of Cherednik algebras of ..."
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Cited by 19 (8 self)
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Double affine Hecke algebras for reduced root systems were introduced by Cherednik [Ch] in order to prove Macdonald conjectures. Double affine Hecke algebras of type C∨Cn were introduced in the works of Noumi, Sahi, and Stokman ([NoSt, Sa, St]) as a generalization of Cherednik algebras of
Naïve Noncommutative Blowups . . .
"... In an earlier paper [KRS] we defined and investigated the properties of the naïve blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one naïvely blows up X at any suitably generic zerodimensional subscheme Z. The resulting alge ..."
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Cited by 18 (9 self)
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In an earlier paper [KRS] we defined and investigated the properties of the naïve blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one naïvely blows up X at any suitably generic zerodimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noetherian and the point modules are never parametrized by a projective scheme. This is despite the fact that the category of torsion modules in qgrA is equivalent to the category of torsion coherent sheaves over X. These results are used in the companion paper [RS1] to prove that a large class of noncommutative surfaces can be written as naïve blowups.
Galgebras, twistings, and equivalences of graded categories
, 2008
"... Given Zgraded rings A and B, we ask when the graded module categories grA and grB are equivalent. Using Zalgebras, we relate the Moritatype results of ÁhnMárki and del Río to the twisting systems introduced by Zhang, and prove, for example: Theorem. If A and B are Zgraded rings, then: (1) A ..."
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Cited by 12 (2 self)
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Given Zgraded rings A and B, we ask when the graded module categories grA and grB are equivalent. Using Zalgebras, we relate the Moritatype results of ÁhnMárki and del Río to the twisting systems introduced by Zhang, and prove, for example: Theorem. If A and B are Zgraded rings, then: (1) A is isomorphic to a Zhang twist of B if and only if the Zalgebras A = L i,j∈Z Aj−i and B = L i,j∈Z Bj−i are isomorphic. (2) If A and B are connected graded with A1̸ = 0, then grA≃grB if and only if A and B are isomorphic. This simplifies and extends Zhang’s results.