<scp>Champ</scp>: a Cherednik algebra<scp>Magma</scp>package

Thiel, Ulrich

doi:10.1112/s1461157015000054

Cited by 19 publications

(28 citation statements)

References 31 publications

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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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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 [73] the author has explicitly computed the solutions to all Problems in Section 2.6 for G 4 and for all parameters using computational methods. In [71] we have shown that for precisely the groups With much more computational effort we have determined in [73] the complete solutions to all problems in Section 2.6 for generic parameters for the groups G 4 , G 5 , G 6 , G 7 , G 8 , G 9 , G 10 , G 12 , G 13 , G 14 , G 15 , G 16 , G 20 , G 22 , G 23 = H 3 , G 24 .…”

Section: Exceptional Groupsmentioning

confidence: 99%

“…We refer to [16] for the details. All results computed so far are available on the author's websites http://www.mathematik.uni-stuttgart.de/~thiel/RRCA and http://thielul.github.io/CHAMP On the latter website the Cherednik Algebra Magma Package CHAMP presented by the author in [73] is freely available. This is a package based on the computer algebra system Magma for performing basic computations in rational Cherednik algebras at arbitrary parameters and in baby Verma modules for restricted rational Cherednik algebras.…”

Section: Exceptional Groupsmentioning

confidence: 99%

Restricted rational Cherednik algebras

Thiel

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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“…This technique has been extended by Geck and Rouquier [14] to a decent class of algebras over integral domains. It became an important tool for studying algebras involving parameters, so for example Hecke algebras (see [13], [12], and [7]) and, more recently, rational Cherednik algebras (see [2], [15], and [28]). We list several further examples in §2A.…”

Section: Introductionmentioning

confidence: 99%

Decomposition Matrices are Generically Trivial

Thiel¹

2015

Int Math Res Notices

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ABSTRACT. We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier of an algebra which is free of finite dimension over a noetherian integral domain and which splits over the fraction field of this ring are generically trivial, i.e., they are trivial in an open neighborhood of the generic point of the spectrum of the base ring. This generalizes a classical result by Brauer in modular representation theory of finite groups. We furthermore show that this is true precisely on an open set in case all fibers of the algebra split. In this way we get a stratification of the base scheme such that decomposition maps are trivial on each stratum. Moreover, this defines a new discriminant of such algebras which generalizes Schur elements of simple modules for symmetric split semisimple algebras. We provide some extensions to the theory of decomposition maps allowing us to work without the usual normality assumption on the base ring.

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Champ: a Cherednik algebraMagmapackage

Cited by 19 publications

References 31 publications

On Hopf algebras with triangular decomposition

On Hopf algebras with triangular decomposition

Restricted rational Cherednik algebras

Decomposition Matrices are Generically Trivial

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