Restricted rational Cherednik algebras

Thiel, Ulrich

doi:10.4171/171-1/24

Cited by 5 publications

(11 citation statements)

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“…From this we easily obtain the proof of Theorem A. We note that the results here together with[42, Appendix B] give a complete description of the representation theory of restricted rational Cherednik algebras for dihedral groups at all parameters. §8A.…”

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confidence: 85%

Cuspidal Calogero–Moser and Lusztig families for Coxeter groups

Bellamy

Thiel

2016

Journal of Algebra

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ABSTRACT. The goal of this paper is to compute the cuspidal Calogero-Moser families for all infinite families of finite Coxeter groups, at all parameters. We do this by first computing the symplectic leaves of the associated Calogero-Moser space and then by classifying certain "rigid" modules. Numerical evidence suggests that there is a very close relationship between Calogero-Moser families and Lusztig families. Our classification shows that, additionally, the cuspidal Calogero-Moser families equal cuspidal Lusztig families for the infinite families of Coxeter groups.

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confidence: 85%

Cuspidal Calogero–Moser and Lusztig families for Coxeter groups

Bellamy

Thiel

2016

Journal of Algebra

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“…The Hilbert series of any finitedimensional Nichols algebra also satisfies this symmetry. The following question for rational Cherednik algebras was posed by Thiel in [37, Question 7.7(a)] and [38,Problem 6.6]; nevertheless some restrictions are required because he also gives counter-examples in the first paper, see [37,Remark 7.8]. In this situation Theorem 2.1 gives us information on the projective u (H,R) (V )modules:…”

Section: On the Representation Theory Of U (Hr) (V )mentioning

confidence: 99%

On Hopf algebras with triangular decomposition

Vay¹

2019

Tensor Categories and Hopf Algebras

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In this survey, we first review some known results on the representation theory of algebras with triangular decomposition, including the classification of the simple modules. We then discuss a recipe to construct Hopf algebras with triangular decomposition. Finally, we extend to these Hopf algebras the main results of [39] regarding projective modules over Drinfeld doubles of bosonizations of Nichols algebras and groups.2010 Mathematics Subject Classification. Primary 16T05, 20G05, 16P10. Partially supported by CONICET, Secyt (UNC), FONCyT PICT 2016-3957 and MathAm-Sud project GR2HOPF.We are interested in graded quotient Hopf algebras of U (H,R) (V ) admitting a triangular decomposition with H in the middle. The main example is the following. Let B(V ) and B(V ) be the Nichols algebras of V and V , with defining ideals J (V ) and J (V ), respectively. We defineProposition 4.2. u (H,R) (V ) is a graded Hopf algebra quotient of U (H,R) (V ) and B(V )⊗H⊗B(V ) −→ u (H,R) (V ) is a triangular decomposition. Also, the Hopf subalgebra generated by V and H, resp. V and H, is isomorphic toProof. We only have to prove the triangular decomposition. A direct computation shows that the composition of the braidings c V,V : V ⊗V → V ⊗V and c V ,V : V ⊗V → V ⊗V is the identity. Then, the Nichols algebra B(V ⊕V ) is isomorphic to B(V )⊗B(V ) as vector spaces by [16, Theorem 2.2Since the generators of J do not belong to this ideal, we have that u (H,R) (V ) ≃ T (V ⊕ V )#H J (V ), J (V ) J ≃ B(V )⊗H⊗B(V ) as vector spaces. 4.1. Examples.

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“…§2D. Gordon's questions These questions are so far only studied for = = C and we cannot go into details about what is already known in this case (see [9, 16.2, 16.4], [10], [17, 6.4, 7.3], [1, §3.3], [25], [26], and [32]). The point is that almost nothing is known for exceptional complex reflection groups and this was one reason for the development of CHAMP.…”

Section: (B) Ifmentioning

confidence: 99%

“…We discussed rational Cherednik algebras for reflection groups over arbitrary fields as long as all reflections are diagonalizable and designed CHAMP to work in this generality. In [32] we computed for example the representation theory of the restricted rational Cherednik algebra attached to the general orthogonal group GO 3 (3) and to modular reflection representations of some symmetric groups. These cases are not yet understood theoretically and we hope that such examples will help to develop a general theory.…”

Section: 2mentioning

confidence: 99%

“…I was partially supported by the DFG Schwerpunktprogramm Darstellungstheorie 1388. We start by reviewing rational Cherednik algebras (see also [9], [17], [5], and [32]) in this paragraph and explain how they can be treated computationally. Instead of the complex numbers as base rings we consider a very general setup here to be able to treat generic parameters algebraically and to introduce analogous problems with modular reflection groups.…”

Section: Introductionmentioning

confidence: 99%

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Champ: a Cherednik algebraMagmapackage

Thiel¹

2015

LMS J. Comput. Math.

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We present a computer algebra package based on Magma for performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded G-module structure of the simple modules, and the Calogero-Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino's conjecture for several exceptional complex reflection groups.Supplementary materials are available with this article.

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Restricted rational Cherednik algebras

Cited by 5 publications

References 51 publications

Cuspidal Calogero–Moser and Lusztig families for Coxeter groups

Cuspidal Calogero–Moser and Lusztig families for Coxeter groups

On Hopf algebras with triangular decomposition

Champ: a Cherednik algebraMagmapackage

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