The ring of twisted tilting modules of quantum $G{L}_n$

Let $G$ be the quantum $G{L}_n$ over a field of characteristic $p\mathrm{\ne }0$. In this paper we define the ring of twisted tilting modules of $G$. We give generators and relations for the ring of twisted tilting modules of quantum $G{L}_2(k)$. We also show that this is a reduced ring