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Embedding of the rank 1 DAHA into Mat(2,Tq) and its automorphisms

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posted on 06.05.2016, 11:04 by Marta Mazzocco
In this review paper we show how the Cherednik algebra of type Č1C1 appears naturally as quantisation of the group algebra of the monodromy group associated to the sixth Painlevé equation. This fact naturally leads to an embedding of the Cherednik algebra of type Č1C1 into Mat(2,Tq), i.e. 2×2 matrices with entries in the quantum torus. For q=1 this result is equivalent to say that the Cherednik algebra of type Č1C1 is Azumaya of degree 2 [31]. By quantising the action of the braid group and of the Okamoto transformations on the monodromy group associated to the sixth Painlevé equation we study the automorphisms of the Cherednik algebra of type Č1C1 and conjecture the existence of a new automorphism. Inspired by the confluences of the Painlevé equations, we produce similar embeddings for the confluent Cherednik algebras HV, HIV, HIII, HII and HI, defined in [27].

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Advanced Studies in Pure Mathematics;

Volume

TBA

Citation

MAZZOCCO, M., 2016. Embedding of the rank 1 DAHA into Mat(2,Tq) and its automorphisms. IN: Konno, H.... et al. (eds.) Representation Theory, Special Functions and Painvlevé Equations (RIMS 2015). Tokoyo: Mathematical Society of Japan, pp. 449-468.

Publisher

Mathematical Society of Japan

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2016

ISBN

9784864970501;9784864970501

Book series

Advanced Studies in Pure Mathematics; 76

Language

en

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