The Quantum Groups \overline{U}_q( sl_2) at the Roots of Unity, Self-Duality and Ascent

Lars Kadison

We compute that $\overline{U}_q(sl_2)$ is self-dual if $q$ is a fourth root of unity. We prove the converse statement by using Radford's formula for the fourth power of the antipode. We end with a discussion of the $\beta$-Frobenius extension structure of $\overline{U}_i(sl_2)$ over the Sweedler $4$-dimensional Hopf subalgebra, and remark that the triangular structure at $\lambda = 0$ ascends for certain general reasons.