On the Smoothness of Centres of Rational Cherednik Algebras in Positive Characteristic

Bellamy, Gwyn; Martino, Maurizio

doi:10.1017/s0017089513000499

Cited by 13 publications

(14 citation statements)

References 37 publications

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“…For a 𝐾-vector space 𝑀 we denote by 𝑀 * := Hom 𝐾 (𝑀, 𝐾) its dual. If 𝑀 ∈ 𝒞(𝐴), then 𝑀 * is naturally an object in 𝒞(𝐴 op ) with grading defined by (8) (…”

Section: Notationmentioning

confidence: 99%

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Highest weight theory for finite-dimensional graded algebras with triangular decomposition

Bellamy

Thiel

2018

Advances in Mathematics

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We consider self-injective finite-dimensional graded algebras admitting a triangular decomposition. In the preceding paper [7] we have shown that the graded module category of such an algebra is a highest weight category and has tilting objects in the sense of Ringel. In this paper we focus on the degree zero part of the algebra, the core of the algebra. We show that the core captures essentially all relevant information about the graded representation theory. Using tilting theory, we show that the core is cellular. We then describe a canonical construction of a highest weight cover, in the sense of Rouquier, of this cellular algebra using a finite subquotient of the highest weight category. Thus, beginning with a self-injective graded algebra admitting a triangular decomposition, we canonically construct a quasi-hereditary algebra which encodes key information, such as the graded multiplicities, of the original algebra. Our results are general and apply to a wide variety of examples, including restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras. We expect that the cell modules and quasi-hereditary algebras introduced here will provide a new way of understanding these important examples.into graded subalgebras given by the multiplication map, where we assume that − is concentrated in negative degree, in degree zero, and + in positive degree.There are a variety of examples of such algebras:(1) Restricted enveloping algebras (g );The proof of this theorem is a modification of very recent work of Andersen-Stroppel-Tubbenhauer [3], who prove this in the case of quantum groups. Unfortunately, one essential ingredient in loc. cit. is the notion of weight spaces, which is

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Section: Notationmentioning

confidence: 99%

“…Restricted rational Cherednik algebras in positive characteristic. We note briefly that restricted rational Cherednik algebras at 𝑡 = 1 in positive characteristic, as studied in [8] 1) ]︁ 𝑊 + ⟩ denotes the 𝑝-coinvariant ring. Here h (1) is the Frobenius twist of h.…”

Section: Triangular Dualitiesmentioning

confidence: 99%

Highest weight theory for finite-dimensional graded algebras with triangular decomposition

Bellamy

Thiel

2018

Advances in Mathematics

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“…Indeed, in the modular setting, the Hochschild cohomology of the acting group algebra is nontrivial and thus gives rise to deformations of the skew group algebra of a completely new flavor. Study has recently begun on the representation theory of deformations of skew group algebras in positive characteristic (see [1,2,3,6,21] for example) and these new types of deformations may help in developing a unified theory. In addition to combinatorial tools, we rely on homological algebra to reveal and to understand these new types of deformations that do not appear in characteristic zero.…”

Section: Introductionmentioning

confidence: 99%

PBW Deformations of Skew Group Algebras in Positive Characteristic

Shepler

Witherspoon

2014

Algebr Represent Theor

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Abstract. We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig's graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeldtype deformation. We fit all these deformations into a general theory, connecting Poincaré-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.

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“…Problem (3) should be a fun exercise even for partial deformations of Kleinian singularities. Problem (5) is important in describing the derived equivalences that are expected between different Q-factorial terminalizations of (h × h * )/Γ.…”

Section: Open Questionsmentioning

confidence: 99%

Hyperplane arrangements associated to symplectic quotient singularities

Bellamy

Schedler

Thiel

2018

Banach Center Publ.

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We study the hyperplane arrangements associated, via the minimal model programme, to symplectic quotient singularities. We show that this hyperplane arrangement equals the arrangement of CM-hyperplanes coming from the representation theory of restricted rational Cherednik algebras. We explain some of the interesting consequences of this identification for the representation theory of restricted rational Cherednik algebras. We also show that the Calogero-Moser space is smooth if and only if the Calogero-Moser families are trivial. We describe the arrangements of CM-hyperplanes associated to several exceptional complex reflection groups, some of which are free.

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On the Smoothness of Centres of Rational Cherednik Algebras in Positive Characteristic

Cited by 13 publications

References 37 publications

Highest weight theory for finite-dimensional graded algebras with triangular decomposition

Highest weight theory for finite-dimensional graded algebras with triangular decomposition

PBW Deformations of Skew Group Algebras in Positive Characteristic

Hyperplane arrangements associated to symplectic quotient singularities

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