We determine the ring structure of the endomorphism algebra of certain tensor powers of modules for the quantum group of $\mathfrak{sl}_2$ in the case where the quantum parameter is allowed to be a root of unity.
In this case there exists - under a suitable localization of our ground ring - a surjection from the group algebra of the braid group to the endomorphism algebra of any tensor power of the Weyl module with highest weight $2$. We take a first step towards determining the kernel of this map by reformulating well-known results on the semisimplicity of the Birman-Murakami-Wenzl algebra in terms of the order of the quantum parameter.
Before we arrive at these main results, we investigate the structure of the endomorphism algebra of the tensor square of any Weyl module.