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.

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

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 $U_q({\widehat{\mathfrak{sl}}}_2)$