Finiteness Conditions, Co-Frobenius Hopf Algebras, and Quantum Groups

Ribbon Element on Co-Frobenius Quasitriangular Hopf Algebras

“…Based on the ideals and results of Beattie, Bulacu and others [7][8][9], we generalize the results of Kauffman and Radford [10] to co-Frobenius quasitriangular Hopf algebras. We find the group-like elements  and g which play a special role in the theory of ribbon Hopf algebras.…”

Ore Extensions of Hopf Group Coalgebras

“…From the point of view of quantum group and Hopf algebra theory, Ore extensions are important for constructing examples of Hopf algebras which are neither commutative nor cocommutative. In recent years, many new examples (often finite dimensional) with special properties were constructed by means of Ore extensions, such as pointed Hopf algebras, co-Frobenius Hopf algebras, and quasitriangular Hopf algebras (see, e.g., [1,2,5]). …”

Co-Frobenius Hopf Algebras: Integrals, Doi–Koppinen Modules and Injective Objects

“…However, the proof was very long and technical. Other proofs have been done recently in [20,3]. We present here a very short coalgebraic proof, which is evidence for the progress done in coalgebra theory during the last two decades.…”

“…Finite dimensional Hopf algebras and cosemisimple Hopf algebras have non-zero integrals. In recent papers [3,2] large classes of other Hopf algebras with non-zero integrals have been constructed. A left integral for the Hopf algebra H is an element t ∈ H * , the dual of H, such that h * t = h * 1 t for any h * ∈ H * .…”