Rational Cherednik Algebras

Gordon, Iain

doi:10.1142/9789814324359_0093

“…We cannot go into details here about the broader context of rational Cherednik algebras and the geometry of Calogero-Moser spaces. There are several extremely useful sources about this and we warmly recommend them to the reader: apart from the original paper by Etingof-Ginzburg [32], there are excellent survey papers by Gordon [42,44] and by Rouquier [65], and excellent lecture notes by Bellamy [5], by Chlouveraki [27], and by Losev [55]. We furthermore highly recommend the wonderful manuscript [13] by Bonnafé and Rouquier. Assumptions.…”

Section: Hecke Algebrasmentioning

confidence: 99%

Restricted rational Cherednik algebras

Thiel

¹

2017

EMS Series of Congress Reports

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Abstract. We give an overview of the representation theory of restricted rational Cherednik algebras. These are certain finite-dimensional quotients of rational Cherednik algebras at t = 0. Their representation theory is connected to the geometry of the Calogero-Moser space, and there is a lot of evidence that they contain certain information about Hecke algebras even though the precise connection is so far unclear. We outline the basic theory along with some open problems and conjectures, and give explicit results in the cyclic and dihedral cases.

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“…Degree-zero deformations of skew group algebras involve parameter maps that identify commutators of elements in V with certain elements of the group algebra. Several important families of these are of broad interest in noncommutative geometry, combinatorics, and representation theory and are the subject of an already extensive literature (see [12] and [13] and further references therein). By comparison, finding elements of degree one with which to identify com-mutators of elements in V requires a more intricate analysis of which cocycles pass obstructions in cohomology in order to determine if the resulting deformations satisfy PBW properties [8,18].…”

Section: Introductionmentioning

confidence: 99%

Degree-One Rational Cherednik Algebras for the Symmetric Group

Foster-Greenwood

¹

,

Kriloff

²

2021

SIGMA

0

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Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of gl n -type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the sl n -type rational Cherednik algebras H 0,c .

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“…Degree-zero deformations of skew group algebras involve parameter maps that identify commutators of elements in V with certain elements of the group algebra. Several important families of these are of broad interest in noncommutative geometry, combinatorics, and representation theory and are the subject of an already extensive literature (see [Gor08] and [Gor10] and further references therein). By comparison, finding elements of degree one with which to identify commutators of elements in V requires a more intricate analysis of which cocycles pass obstructions in cohomology in order to determine if the resulting deformations satisfy PBW properties [SW12,FGK17].…”

Section: Introductionmentioning

confidence: 99%

Degree-One Rational Cherednik Algebras for the Symmetric Group

Foster-Greenwood,

Kriloff

2019

Preprint

0

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Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of gl n -type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the sln-type rational Cherednik algebras H0,c.

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Rational Cherednik Algebras

Abstract: Abstract. We survey a number of results about rational Cherednik algebra representation theory and its connection to symplectic singularities and their resolutions.

Cited by 9 publications

References 64 publications

Restricted rational Cherednik algebras

Restricted rational Cherednik algebras

Degree-One Rational Cherednik Algebras for the Symmetric Group

Degree-One Rational Cherednik Algebras for the Symmetric Group

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