On the structure of $End_{u_k(2)}(\Omega_k^{\otimes r})$

Abstract

Let $u_k(2)$ be the infinitesimal quantum $\frak{gl}_2$ over $k$, where $k$ is a field containing an $l$th primitive root $\epsilon$ of 1 with $l\geq 3$ {\it odd}. We will determine the basic algebra for ${u_k(2)}(\Omega_k^{\otimes r})$, where $\Omega_k$ is the natural module for $u_k(2)$.