## Advanced Studies in Pure Mathematics

### Elliptic quantum groups $U_{q,p}(\widehat{\mathfrak{gl}}_N)$ and $E_{q,p}(\widehat{\mathfrak{gl}}_N)$

Hitoshi Konno

#### Abstract

We reformulate a central extension of Felder's elliptic quantum group in the FRST formulation as a topological algebra $E_{q,p}(\widehat{\mathfrak{gl}}_N)$ over the ring of formal power series in $p$. We then discuss the isomorphism between $E_{q,p}(\widehat{\mathfrak{gl}}_N)$ and the elliptic algebra $U_{q,p}(\widehat{\mathfrak{gl}}_N)$ of the Drinfeld realization. An evaluation $H$-algebra homomorphism from $U_{q,p}(\widehat{\mathfrak{gl}}_N)$ to a dynamical extension of the quantum affine algebra $U_{q}(\widehat{\mathfrak{gl}}_N)$ resolves the problem into the one discussed by Ding and Frenkel in the trigonometric case. We also provide some useful formulas for the elliptic quantum determinants.

#### Article information

Dates
Revised: 5 September 2016
First available in Project Euclid: 21 September 2018

https://projecteuclid.org/ euclid.aspm/1537499431

Digital Object Identifier
doi:10.2969/aspm/07610347

Mathematical Reviews number (MathSciNet)
MR3837927

Zentralblatt MATH identifier
07039308

#### Citation

Konno, Hitoshi. Elliptic quantum groups $U_{q,p}(\widehat{\mathfrak{gl}}_N)$ and $E_{q,p}(\widehat{\mathfrak{gl}}_N)$. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 347--417, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610347. https://projecteuclid.org/euclid.aspm/1537499431