The algebra $U_q^{+}$ and its alternating central extension ${\mathcal{U}}_q^{+}$

Abstract

Let $U_q^{+}$ denote the positive part of the quantized enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$. The algebra $U_q^{+}$ has a presentation involving two generators $W_0$, $W_1$ and two relations, called the $q$-Serre relations. In 1993 I. Damiani obtained a PBW basis for $U_q^{+}$, consisting of some elements $\{E_{n\delta +{\alpha }_0}{\}}_{n=0}^{\infty }$, $\{E_{n\delta +{\alpha }_1}{\}}_{n=0}^{\infty }$, $\{E_{n\delta }{\}}_{n=1}^{\infty }$. In 2019 we introduced the alternating central extension ${\mathcal{U}}_q^{+}$ of $U_q^{+}$. We defined ${\mathcal{U}}_q^{+}$ by generators and relations. The generators, said to be alternating, are denoted $\{{\mathcal{W}}_{-k}{\}}_{k=0}^{\infty }$, $\{{\mathcal{W}}_{k+1}{\}}_{k=0}^{\infty }$, $\{{\mathcal{G}}_{k+1}{\}}_{k=0}^{\infty }$, $\{{\tilde{\mathcal{G}}}_{k+1}{\}}_{k=0}^{\infty }$. Let $⟨{\mathcal{W}}_0,{\mathcal{W}}_1⟩$ denote the subalgebra of ${\mathcal{U}}_q^{+}$ generated by ${\mathcal{W}}_0$, ${\mathcal{W}}_1$. It is known that there exists an algebra isomorphism $U_q^{+}\to ⟨{\mathcal{W}}_0,{\mathcal{W}}_1⟩$ that sends $W_0\mapsto {\mathcal{W}}_0$ and $W_1\mapsto {\mathcal{W}}_1$. Via this isomorphism we identify $U_q^{+}$ with $⟨{\mathcal{W}}_0,{\mathcal{W}}_1⟩$. In our main result, we express the Damiani PBW basis elements in terms of the alternating generators. We give the answer in terms of generating functions.

arXiv comment22 pages. arXiv admin note: text overlap with arXiv:2106.14041

References in corpus (2)

• A new (in)finite dimensional algebra for quantum integrable models
• A new current algebra and the reflection equation

Cited by in corpus (2)

• 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 $U_q({\widehat{\mathfrak{sl}}}_2)$