On the second realization for the positive part of $U_q(\widehat{s{l}_2})$ of equitable type

Abstract

The equitable presentation of the quantum algebra $U_q(\widehat{s{l}_2})$ 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 $U_q(\widehat{s{l}_2})$ of equitable type $U_q^{IT,+}$ 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 $U_q(\widehat{s{l}_2})$ 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 $U_q(\widehat{g{l}_2})$ is also described using the framework of Freidel-Maillet algebras.

arXiv comment19 pages; v2: Lemma 3.3 added

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Cited by in corpus (2)

• Using Catalan words and a $q$-shuffle algebra to describe the Beck PBW basis for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
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