The group of bi-Galois objects over the coordinate algebra of the Frobenius-Lusztig kernel of \(\mathrm{SL}(2)\) (Q2816790)

Assume that \(q\) is a root of unity of odd order \(N>1\) and \(u_q(sl(2))\) is the Frobenius-Lusztig kernel of \(\mathrm{SL}(2)\). The author wants to compute the the group of bi-Galois objects over \(u_q(sl(2))^*\), which is the coordinate algebra of \(u_q(sl(2))\). It is obtained that there exists an injective group morphism \(\mathrm{PSL}(n)\to \text{BiGal}(u_q(sl(n))^*)\) for \(n\geq 2\), which is an isomorphism for \(n=2\). As a consequence, it is shown that the lazy cohomology of \(u_q(sl(2))^*\) is trivial. It should be noted that there is an important difference between the case of \(u_q(sl(2))^*\) and the few other known results of finite dimensional non-cosemisimple Hopf algebras is that \(u_q(sl(2))^*\) is not pointed.