A generating function associated with the alternating elements in the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$

Abstract

The positive part $U_q^{+}$ of $U_q({\widehat{\mathfrak{sl}}}_2)$ admits an embedding into a $q$-shuffle algebra. This embedding was introduced by M. Rosso in 1995. In 2019, Terwilliger introduced the alternating elements $\{W_{-n}{\}}_{n\in \mathbb{N}}$, $\{W_{n+1}{\}}_{n\in \mathbb{N}}$, $\{G_{n+1}{\}}_{n\in \mathbb{N}}$, $\{{\tilde{G}}_{n+1}{\}}_{n\in \mathbb{N}}$ in $U_q^{+}$ using the Rosso embedding. He showed that the alternating elements $\{W_{-n}{\}}_{n\in \mathbb{N}}$, $\{W_{n+1}{\}}_{n\in \mathbb{N}}$, $\{{\tilde{G}}_{n+1}{\}}_{n\in \mathbb{N}}$ form a PBW basis for $U_q^{+}$, and he expressed $\{G_{n+1}{\}}_{n\in \mathbb{N}}$ in this alternating PBW basis. In his calculation, Terwilliger used some elements $\{D_n{\}}_{n\in \mathbb{N}}$ with the following property: the generating function $D(t)={\sum }_{n\in \mathbb{N}}{D}_n{t}^n$ is the multiplicative inverse of the generating function $\tilde{G}(t)={\sum }_{n\in \mathbb{N}}{\tilde{G}}_n{t}^n$ where ${\tilde{G}}_0=1$. Terwilliger defined $\{D_n{\}}_{n\in \mathbb{N}}$ recursively; in this paper, we will express $\{D_n{\}}_{n\in \mathbb{N}}$ in closed form.

16 pages; minor changes in abstract and introduction; added two propositions in section 3; minor adjustment on bibliography; fixed a typo

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