Abstract
This paper concerns the positive part Uq+ of the quantum group Uq(sl^2). The algebra Uq+ has a presentation involving two generators that satisfy the cubic q-Serre relations. We recently introduced an algebra Uq+ called the alternating central extension of Uq+. We presented Uq+ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of Uq+ that involves a small subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of Uq+.
Highlights
The algebra Uq(sl2) is well known in representation theory [15] and statistical mechanics [21]
We denote by U the algebra presented in Definition 3.1, and eventually prove that U = Uq+. After this result is established, we describe some features of Uq+ that are illuminated by the presentation in Definition 3.1
We prove that U = Uq+
Summary
The algebra Uq(sl2) is well known in representation theory [15] and statistical mechanics [21]. By [30, Lemma 8.4] and the previous comments, the algebra Uq+ is presented by its alternating elements and the relations in [30, Propositions 5.7, 5.10, 8.1]. For this presentation it is natural to ask what happens if the relations in [30, Proposition 8.1] are removed To answer this question, in [31, Definition 3.1] we defined an algebra Uq+ by generators and relations in the following way. We mentioned above that the algebra Uq+ is presented by its alternating generators and the relations in [30, Propositions 5.7, 5.10]. Using this result, we prove that U = Uq+. For the elements (2.2) of Uq+, the same notation will be used for their -images in W0, W1
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