paper

On the second realization for the positive part of $U_{q} (s l_{2})$ of equitable type

arXiv:2106.11706 · doi:10.1007/s11005-021-01502-1

Abstract

The equitable presentation of the quantum algebra $U_{q} (s l_{2})$ is considered. This presentation was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works the (nonstandard) positive part of $U_{q} (s l_{2})$ of equitable type $U_{q}^{I T, +}$ and its second realization (current algebra) $U_{q}^{T, +}$ are introduced and studied. A presentation for $U_{q}^{T, +}$ is given in terms of a K-operator satisfying a Freidel-Maillet type equation and a condition on its quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel L-operators are considered, from which an explicit injective homomorphism from $U_{q}^{T, +}$ to a subalgebra of Drinfeld's second realization (current algebra) of $U_{q} (s l_{2})$ is derived, and the comodule algebra structure of $U_{q}^{T, +}$ is characterized. The central extension of $U_{q}^{T, +}$ and its relation with Drinfeld's second realization of $U_{q} (g l_{2})$ is also described using the framework of Freidel-Maillet algebras.

19 pages; v2: Lemma 3.3 added

On the second realization for the positive part of Uq​(sl2​​) of equitable type

Abstract

On the second realization for the positive part of $U_{q} (s l_{2})$ of equitable type