A compact presentation for the alternating central extension of the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$

Abstract

This paper concerns the positive part $U_q^{+}$ of the quantum group $U_q({\widehat{\mathfrak{sl}}}_2)$. The algebra $U_q^{+}$ has a presentation involving two generators that satisfy the cubic $q$-Serre relations. We recently introduced an algebra ${\mathcal{U}}_q^{+}$ called the alternating central extension of $U_q^{+}$. We presented ${\mathcal{U}}_q^{+}$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of ${\mathcal{U}}_q^{+}$ that involves a small subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of ${\mathcal{U}}_q^{+}$.

arXiv comment23 pages

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Cited by in corpus (4)

• The $q$-Onsager algebra and its alternating central extension
• Tridiagonal pairs, alternating elements, and distance-regular graphs
• A generating function associated with the alternating elements in the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Using a $q$-shuffle algebra to describe the basic module $V({\Lambda }_0)$ for the quantized enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$