THE QUANTUM GROUPS \overline{U}_{q}(sl_{2}) AT THE ROOTS OF UNITY, SELF-DUALITY AND ASCENT

LARS KADISON

Abstract. 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 β-Frobenius extension structure of \overline{U}_{i}(sl_{2}) over the Sweedler 4-dimensional Hopf subalgebra, and remark that the triangular structure at λ = 0 ascends for certain general reasons.