### Abstract

Original language | English |
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Publisher | ArXiv |

Publication status | Submitted - 26 Jun 2015 |

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### Keywords

- quantum group

### Cite this

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

Research output: Working paper

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TY - UNPB

T1 - A proof of De Concini-Kac-Procesi conjecture I. Representations of quantum groups at roots of unity and q-W algebras

AU - Sevastyanov, Alexey

N1 - 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.

PY - 2015/6/26

Y1 - 2015/6/26

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.

KW - quantum group

M3 - Working paper

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

PB - ArXiv

ER -