## 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), 95 - 123

### The hyperbolic modular double and the Yang-Baxter equation

Dmitry Chicherin and Vyacheslav P. Spiridonov

#### Abstract

We construct a hyperbolic modular double – an algebra lying in between the Faddeev modular double for $U_q(sl_2)$ and the elliptic modular double. The intertwining operator for this algebra leads to an integral operator solution of the Yang-Baxter equation associated with a generalized Faddeev-Volkov lattice model introduced by the second author. We describe also the L-operator and finite-dimensional R-matrices for this model.

#### Article information

**Dates**

Received: 1 November 2015

First available in Project Euclid:
21 September 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1537499424

**Digital Object Identifier**

doi:10.2969/aspm/07610095

**Mathematical Reviews number (MathSciNet)**

MR3837920

**Zentralblatt MATH identifier**

07039301

**Subjects**

Primary: 33D60: Basic hypergeometric integrals and functions defined by them 39A13: Difference equations, scaling ($q$-differences) [See also 33Dxx] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

**Keywords**

Yang-Baxter equation Faddeev-Volkov model Sklyanin algebra modular double solvable lattice models

#### Citation

Chicherin, Dmitry; Spiridonov, Vyacheslav P. The hyperbolic modular double and the Yang-Baxter equation. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 95--123, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610095. https://projecteuclid.org/euclid.aspm/1537499424