The alternating central extension for the positive part of Uq(slˆ2)

Abstract

This paper is about the positive part Uq+ of the quantum group Uq(slˆ2). The algebra Uq+ has a presentation with two generators A,B that satisfy the cubic q-Serre relations. Recently we introduced a type of element in Uq+, said to be alternating. Each alternating element commutes with exactly one of A, B, qBA−q−1AB, qAB−q−1BA; this gives four types of alternating elements. There are infinitely many alternating elements of each type, and these mutually commute. In the present paper we use the alternating elements to obtain a central extension U+q of Uq+. We define Uq+ by generators and relations. These generators, said to be alternating, are in bijection with the alternating elements of Uq+. We display a surjective algebra homomorphism Uq+→Uq+ that sends each alternating generator of Uq+ to the corresponding alternating element in Uq+. We adjust this homomorphism to obtain an algebra isomorphism Uq+→Uq+⊗F[z1,z2,…] where F is the ground field and {zn}n=1∞ are mutually commuting indeterminates. We show that the alternating generators form a PBW basis for Uq+. We discuss how Uq+ is related to the work of Baseilhac, Koizumi, Shigechi concerning the q-Onsager algebra and integrable lattice models.