$U_q^{+}(B_2)$ and its representations

Abstract

In this article we investigate the algebra $U_q^{+}(B_2)$. Assume that $q$ is a primitive $m$-th root of unity with $m\ge 5$. 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 $U_q^{+}(B_2)$.

Preliminary Version. Comments are welcomed