On pointed ribbon Hopf algebras
This paper deals mainly with the families of pointed Hopf algebras constructed by Radford: \(H_{n,q,N,\nu}\), which generalize Sweedler's 4-dimensional noncommutative non-cocommutative Hopf algebra, and \(U_{(N,\nu,\omega)}\), which is a family of finite dimensional pointed unimodular ribbon Hopf algebras, and which generalizes the well known quantum group \(U_q(\text{sl}_2)'\) (\(q\) is a root of unity.) The paper focusses on three aspects of these families of Hopf algebras. First, the author proves the uniqueness of these families as quasitriangular Hopf algebras, with respect to certain properties of the grouplikes and the skew primitives. Next, he finds all the possible quasitriangular structures (up to equivalence) of \(U_{(N,\nu,\omega)}\) and shows that they are all ribbon; in the process he describes its group of automorphisms and all its Hopf subalgebras. In the last section, he applies these results in two contexts. First he applies them to the quantum group \(U_q(\text{sl}_2)'\) (\(q\) a root of unity), giving, for example, all possible \(R\)-matrices that render it quasitriangular. Then he uses the 8-dimensional \(U_{(2,1,-1)}\) to compute invariants of lens spaces; it turns out that this Hopf algebra distinguishes between any two lens spaces, and it also shows that Hennings' 3-manifold invariant is not trivial.
scientific article; zbMATH DE number 883785