## 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), 155 - 193

### On connection matrices of quantum Knizhnik-Zamolodchikov equations based on Lie super algebras

Wellington Galleas and Jasper V. Stokman

#### Abstract

We propose a new method to compute connection matrices of quantum Knizhnik-Zamolodchikov equations associated to integrable vertex models with super algebra and Hecke algebra symmetries. The scheme relies on decomposing the underlying spin representation of the affine Hecke algebra in principal series modules and invoking the known solution of the connection problem for quantum affine Knizhnik-Zamolodchikov equations associated to principal series modules. We apply the method to the spin representation underlying the $\mathcal{U}_q\bigl(\widehat{\mathfrak{gl}}(2|1)\bigr)$ Perk-Schultz model. We show that the corresponding connection matrices are described by an elliptic solution of the dynamical quantum Yang-Baxter equation with spectral parameter.

#### Article information

**Dates**

Received: 15 October 2015

Revised: 25 March 2016

First available in Project Euclid:
21 September 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1537499426

**Digital Object Identifier**

doi:10.2969/aspm/07610155

**Mathematical Reviews number (MathSciNet)**

MR3837922

**Zentralblatt MATH identifier**

07039303

**Subjects**

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 17B80: Applications to integrable systems 20C08: Hecke algebras and their representations 81R05: Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70]

**Keywords**

Quantum Knizhnik-Zamolodchikov equations dynamical quantum Yang-Baxter equation affine Hecke algebras quantum super algebras connection problem

#### Citation

Galleas, Wellington; Stokman, Jasper V. On connection matrices of quantum Knizhnik-Zamolodchikov equations based on Lie super algebras. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 155--193, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610155. https://projecteuclid.org/euclid.aspm/1537499426