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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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

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

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

Permanent link to this document euclid.aspm/1537499431

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Elliptic quantum group Hopf algebroid quantum determinant


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.

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