Deformed preprojective algebras and symplectic reflection algebras for wreath products

Gan, Wee Liang; Ginzburg, Victor

doi:10.1016/j.jalgebra.2004.08.007

Cited by 20 publications

(30 citation statements)

References 7 publications

(11 reference statements)

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“…Results in [3] then imply that H is also closely related to several other algebras appearing in the literature, e.g. H is the spherical subalgebra of a certain deformed preprojective algebra by Gan and Ginzburg [7]. In this paper we obtain several new properties of the algebra H. One remarkable property is that it contains a subalgebra H that may be considered as a deformation of C[W ] with deformed braid-relations.…”

Section: Introductionmentioning

confidence: 57%

Multivariable Wilson polynomials and degenerate Hecke algebras

Groenevelt

2009

Sel. Math. New Ser.

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Abstract. We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type (C ∨ n , Cn). A certain representation in terms of difference-reflection operators naturally leads to the definition of nonsymmetric versions of the multivariable Wilson polynomials. Using the degenerate Hecke algebra we derive several properties, such as orthogonality relations and quadratic norms, for the nonsymmetric and symmetric multivariable Wilson polynomials. Mathematics Subject Classification (2000). Primary 33C52, 33C80; Secondary 20C08.

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Section: Introductionmentioning

confidence: 57%

Multivariable Wilson polynomials and degenerate Hecke algebras

Groenevelt

2009

Sel. Math. New Ser.

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“…Quiver Lie algebras. Symplectic reflection algebras for wreath products of G are known to be Morita equivalent to certain deformed preprojective algebras of affine Dynkin quivers, which are called Gan-Ginzburg algebras in the literature [Gan and Ginzburg 2005]. In the rank one case, these are the usual deformed preprojective algebras λ (Q).…”

Section: Further Discussionmentioning

confidence: 99%

Double affine Lie algebras and finite groups

Guay¹,

Hernandez²,

Loktev³

2009

Pacific J. Math.

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We begin to study the Lie theoretical analogues of symplectic reflection algebras for a finite cyclic group ; we call these algebras "cyclic double affine Lie algebras". We focus on type A: In the finite (respectively affine, double affine) case, we prove that these structures are finite (respectively affine, toroidal) type Lie algebras, but the gradings differ. The case that is essentially new is sl n (‫ [ރ‬u, v] ). We describe its universal central extensions and start the study of its representation theory, in particular of its highest weight integrable modules and Weyl modules. We also consider the first Weyl algebra A 1 instead of the polynomial ring ‫ [ރ‬u, v], and, more generally, a rank one rational Cherednik algebra. We study quasifinite highest weight representations of these Lie algebras.

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“…Definition 6.1 ( [12]). Let λ = (λ i ) i∈I ∈ C ⊕|I| and ν ∈ C. The deformed preprojective algebra Π λ,ν l (Q) (also called Gan-Ginzburg algebra in the literature) is defined as the quotient of T B E S l by the following relations:…”

Section: Gan-ginzburg Algebrasmentioning

confidence: 99%

“…We are able to generalize to D λ,ν n (Q) some of the main results of [4,11,12]. When the graph underlying Q is an affine Dynkin diagram of type A, D or E corresponding to a finite subgroup Γ of SL 2 (C) via the McKay correspondence, we connect a certain subalgebra of the Γ-DDCA D β,b n (Γ) to a quotient of D λ,ν n (Q).…”

Section: Introductionmentioning

confidence: 96%

“…The theory of symplectic reflection algebras introduced by P. Etingof and V. Ginzburg in [10] is a generalization of the Crawley-Boevey-Holland theory of non-commutative deformations. In [12], symplectic reflection algebras for wreath products of finite subgroups Γ of SL 2 (C) were shown to be Morita equivalent to a new family of algebras Π λ,ν l (Q) which can be seen as deformed preprojective algebras for wreath products S l Γ; they are also called Gan-Ginzburg algebras in the literature. (We denote by S l the symmetric group on l letters.…”

Section: Introductionmentioning

confidence: 99%

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Quantum algebras and quivers

Guay

2009

Sel. math., New ser.

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Given a finite quiver Q without loops, we introduce a new class of quantum algebras D(Q) which are deformations of the enveloping algebra of a Lie algebra which is a central extension of sln(Π(Q)) where Π(Q) is the preprojective algebra of Q. When Q is an affine Dynkin quiver of type A, D or E, we can relate them to Γ-deformed double current algebras. We are able to construct functors between different categories of modules over D(Q). We also give some general results about b sln(A) for a quadratic algebra A and about b g(C [u, v]), which we use to introduce deformed double current algebras associated to a simple Lie algebra g. (2000). Primary 17B37; Secondary 17B65. Mathematics Subject Classification

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Deformed preprojective algebras and symplectic reflection algebras for wreath products

Cited by 20 publications

References 7 publications

Multivariable Wilson polynomials and degenerate Hecke algebras

Multivariable Wilson polynomials and degenerate Hecke algebras

Double affine Lie algebras and finite groups

Quantum algebras and quivers

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