The algebra Uq+ and its alternating central extension 𝒰q+

Abstract

Let [Formula: see text] denote the positive part of the quantized enveloping algebra [Formula: see text]. The algebra [Formula: see text] has a presentation involving two generators [Formula: see text], [Formula: see text] and two relations, called the [Formula: see text]-Serre relations. In 1993 Damiani obtained a PBW basis for [Formula: see text], consisting of some elements [Formula: see text], [Formula: see text], [Formula: see text]. In 2019, we introduced the alternating central extension [Formula: see text] of [Formula: see text]. We defined [Formula: see text] by generators and relations. The generators, said to be alternating, are denoted [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. Let [Formula: see text] denote the subalgebra of [Formula: see text] generated by [Formula: see text], [Formula: see text]. It is known that there exists an algebra isomorphism [Formula: see text] that sends [Formula: see text] and [Formula: see text]. Via this isomorphism we identify [Formula: see text] with [Formula: see text]. 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. A key step is a factorization of the generating function for the elements that generate the center of [Formula: see text].