A generating function associated with the alternating elements in the positive part of Uq(sl^2). Issue 4 (3rd April 2023) - Ruan, Chenwei
Abstract: The positive part U q + of U q ( s l ̂ 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 ∈ N,   { W n + 1 } n ∈ N, { G n + 1 } n ∈ N,   { G ˜ n + 1 } n ∈ N in U q + using the Rosso embedding. He showed that the alternating elements { W − n } n ∈ N,   { W n + 1 } n ∈ N,   { G ˜ n + 1 } n ∈ N form a PBW basis for U q +, and he expressed { G n + 1 } n ∈ N in this alternating PBW basis. In his calculation, Terwilliger used some elements { D n } n ∈ N with the following property: the generating function D ( t ) = ∑ n ∈ N D n t n is the multiplicative inverse of the generating function G ˜ ( t ) = ∑ n ∈ N G ˜ n t n where G ˜ 0 = 1 . Terwilliger defined { D n } n ∈ N recursively; in this paper, we will express { D n } n ∈ N in closed form. Communicated by Alberto Elduque