On the structure of $En{d}_{u_k(2)}(\hat{I}{\text{c◯}}_k^{\otimes r})$

Abstract

Let $u_k(2)$ be the infinitesimal quantum ${\mathfrak{gl}}_2$ over $k$, where $k$ is a field containing an $l$th primitive root $\hat{I}\mu$ of 1 with $l\ge 3$ {\it odd}. We will determine the basic algebra for $u_k(2)(\hat{I}{\text{c◯}}_k^{\otimes r})$, where $\hat{I}{\text{c◯}}_k$ is the natural module for $u_k(2)$.