  We define notions of pivotal and ribbon objects in a monoidal category. These
constructions give pivotal or ribbon monoidal categories from a monoidal
category which is not necessarily with duals. We apply this construction to the
braided monoidal category of Yetter--Drinfeld modules over a Hopf algebra. This
gives rise to the notion of ribbon Yetter--Drinfeld modules over a Hopf
algebra, which form ribbon categories. This gives an invariant of tangles.

Quantum Algebra;Category Theory