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

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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

Source
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), 347-417

Dates
Received: 14 March 2016
Revised: 5 September 2016
First available in Project Euclid: 21 September 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1537499431

Digital Object Identifier
doi:10.2969/aspm/07610347

Mathematical Reviews number (MathSciNet)
MR3837927

Zentralblatt MATH identifier
07039308

Subjects
Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50] 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations [See also 17B65, 17B67, 22E65, 22E67, 22E70] 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]

Keywords
Elliptic quantum group Hopf algebroid quantum determinant

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


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