A proof of De Concini-Kac-Procesi conjecture I. Representations of quantum groups at roots of unity and q-W algebras

Research output: Working paper

Abstract

Let U_q(g) be the standard simply connected version of the Drinfeld-Jumbo quantum group at an odd m-th root of unity q. De Concini, Kac and Proicesi observed that isomorphism classes of irreducible representations of U_q(g) are parameterized by the conjugacy classes in the connected simply connected algebraic group G corresponding to the simple complex Lie algebra g. They also conjectured that the dimension of a representation corresponding to a conjugacy class O is divisible by m^{1/2 dim O}. We show that if O intersects one of special transversal slices S to the set of conjugacy classes in G defined in arXiv:0809.0205 then the dimension of every finite-dimensional irreducible representation of U_q(g) corresponding to O is divisible by m^{1/2 codim S}. In the second part of this paper arXiv:1403.4108 is shown that for every conjugacy class O in G one can find a transversal slice S such that O intersects S and dim O = codim S. This proves the De Concini-Kac-Procesi conjecture. Our result also implies an equivalence between a category of finite-dimensional U_q(g)-modules and a category of finite-dimensional representations of a q-W algebra which can be regarded as a truncation of the quantized algebra of regular functions on S.
Original languageEnglish
PublisherArXiv
Publication statusSubmitted - 26 Jun 2015

Fingerprint

W-algebras
Roots of Unity
Conjugacy class
Quantum Groups
Divisible
Intersect
Irreducible Representation
Slice
Isomorphism Classes
Algebraic Groups
Truncation
Lie Algebra
Odd
Equivalence
Imply
Module
Algebra

Keywords

  • quantum group

Cite this

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title = "A proof of De Concini-Kac-Procesi conjecture I. Representations of quantum groups at roots of unity and q-W algebras",
abstract = "Let U_q(g) be the standard simply connected version of the Drinfeld-Jumbo quantum group at an odd m-th root of unity q. De Concini, Kac and Proicesi observed that isomorphism classes of irreducible representations of U_q(g) are parameterized by the conjugacy classes in the connected simply connected algebraic group G corresponding to the simple complex Lie algebra g. They also conjectured that the dimension of a representation corresponding to a conjugacy class O is divisible by m^{1/2 dim O}. We show that if O intersects one of special transversal slices S to the set of conjugacy classes in G defined in arXiv:0809.0205 then the dimension of every finite-dimensional irreducible representation of U_q(g) corresponding to O is divisible by m^{1/2 codim S}. In the second part of this paper arXiv:1403.4108 is shown that for every conjugacy class O in G one can find a transversal slice S such that O intersects S and dim O = codim S. This proves the De Concini-Kac-Procesi conjecture. Our result also implies an equivalence between a category of finite-dimensional U_q(g)-modules and a category of finite-dimensional representations of a q-W algebra which can be regarded as a truncation of the quantized algebra of regular functions on S.",
keywords = "quantum group",
author = "Alexey Sevastyanov",
note = "The author is grateful to Giovanna Carnovale, Iulian Ion Simion, Lew is Topley and to the members of representation theory seminars at the Universities of Bologna and Padua for careful reading of the manuscript.",
year = "2015",
month = "6",
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language = "English",
publisher = "ArXiv",
type = "WorkingPaper",
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N2 - Let U_q(g) be the standard simply connected version of the Drinfeld-Jumbo quantum group at an odd m-th root of unity q. De Concini, Kac and Proicesi observed that isomorphism classes of irreducible representations of U_q(g) are parameterized by the conjugacy classes in the connected simply connected algebraic group G corresponding to the simple complex Lie algebra g. They also conjectured that the dimension of a representation corresponding to a conjugacy class O is divisible by m^{1/2 dim O}. We show that if O intersects one of special transversal slices S to the set of conjugacy classes in G defined in arXiv:0809.0205 then the dimension of every finite-dimensional irreducible representation of U_q(g) corresponding to O is divisible by m^{1/2 codim S}. In the second part of this paper arXiv:1403.4108 is shown that for every conjugacy class O in G one can find a transversal slice S such that O intersects S and dim O = codim S. This proves the De Concini-Kac-Procesi conjecture. Our result also implies an equivalence between a category of finite-dimensional U_q(g)-modules and a category of finite-dimensional representations of a q-W algebra which can be regarded as a truncation of the quantized algebra of regular functions on S.

AB - Let U_q(g) be the standard simply connected version of the Drinfeld-Jumbo quantum group at an odd m-th root of unity q. De Concini, Kac and Proicesi observed that isomorphism classes of irreducible representations of U_q(g) are parameterized by the conjugacy classes in the connected simply connected algebraic group G corresponding to the simple complex Lie algebra g. They also conjectured that the dimension of a representation corresponding to a conjugacy class O is divisible by m^{1/2 dim O}. We show that if O intersects one of special transversal slices S to the set of conjugacy classes in G defined in arXiv:0809.0205 then the dimension of every finite-dimensional irreducible representation of U_q(g) corresponding to O is divisible by m^{1/2 codim S}. In the second part of this paper arXiv:1403.4108 is shown that for every conjugacy class O in G one can find a transversal slice S such that O intersects S and dim O = codim S. This proves the De Concini-Kac-Procesi conjecture. Our result also implies an equivalence between a category of finite-dimensional U_q(g)-modules and a category of finite-dimensional representations of a q-W algebra which can be regarded as a truncation of the quantized algebra of regular functions on S.

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