## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, H. Konno, H. Sakai, J. Shiraishi, T. Suzuki and Y. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2018), 449 - 468

### Embedding of the rank 1 DAHA into $Mat(2,\mathbb T_q)$ and its automorphisms

#### Abstract

In this paper we show how the Cherednik algebra of type $\check{C_1}C_1$ 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 $\check{C_1}C_1$ into $Mat(2,\mathbb T_q)$, i.e. $2\times 2$ matrices with entries in the quantum torus. For $q=1$ this result is equivalent to say that the Cherednik algebra of type $\check{C_1}C_1$ 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 $\check{C_1}C_1$ 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 $\mathcal H_V,\mathcal H_{IV},\mathcal H_{III},\mathcal H_{II}$ and $\mathcal H_{I},$ defined in [27].

#### Article information

**Dates**

Received: 12 December 2015

Revised: 10 March 2016

First available in Project Euclid:
21 September 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1537499433

**Digital Object Identifier**

doi:10.2969/aspm/07610449

**Mathematical Reviews number (MathSciNet)**

MR3837929

**Zentralblatt MATH identifier**

07039310

**Subjects**

Primary: 33D80: Connections with quantum groups, Chevalley groups, $p$-adic groups, Hecke algebras, and related topics 33D52: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) 16T99: None of the above, but in this section

**Keywords**

Double Affine Hecke Algebra Monodromy preserving deformations Painlevé equations

#### Citation

Mazzocco, Marta. Embedding of the rank 1 DAHA into $Mat(2,\mathbb T_q)$ and its automorphisms. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 449--468, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610449. https://projecteuclid.org/euclid.aspm/1537499433