$U_q^+(B_2)$ and its representations

TL;DR

This work analyzes $U_q^+(B_2)$ at a primitive root of unity, establishing its PI structure and determining its center. By introducing new generators $e_3$ and $z$, and constructing a 'good' subalgebra $ extbf{B}$ as a generalized Weyl algebra, the authors reduce the classification of simple modules to manageable subproblems, splitting into $e_3$-torsionfree and nilpotent cases. They completely classify simple modules in both regimes, provide explicit module families with parameterizations, and derive explicit isomorphism criteria within each family. The center is computed, revealing generators $e_1^l,e_2^l,e_3^l,z,z_1$ with a central element $z_1=e_1\tilde z+\frac{e_3^2}{q^4-1}$, and the PI-degree is shown to be $l=\mathrm{ord}(q^2)$, linking representation theory to the PI structure.

Abstract

In this article we investigate the algebra $U_q^+(B_2)$. Assume that $q$ is a primitive $m$-th root of unity with $m \geq 5$. We prove that $U_q^+(B_2)$ becomes a Polynomial Identity (PI) algebra. It was previously known that for such algebras the simple modules are finite-dimensional with dimension at most the PI degree. We determine the PI degree of $U_q^+(B_2)$ and we classify up to isomorphism the simple $U_q^+(B_2)$-modules. We also find the center of $U_q^+(B_2)$.

Theorems & Definitions (42)

• Lemma 2.1
• proof
• Corollary 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• Remark 2.5
• Proposition 2.6
• Theorem 2.7
• Definition 3.1