Ribbon Yetter--Drinfeld modules and tangle invariants

TL;DR

The paper introduces ribbon Yetter--Drinfeld modules over Hopf algebras, constructing ribbon categories that yield new tangle invariants by extending monoidal categories with pivotal and ribbon structures.

Contribution

It develops a novel framework for ribbon categories from Yetter--Drinfeld modules, enabling the creation of new tangle invariants without requiring duals in the original category.

Findings

Defined pivotal and ribbon objects in monoidal categories.

Constructed ribbon categories from Yetter--Drinfeld modules.

Provided a new invariant of tangles.

Abstract

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.