Ribbon category

introduction

important class of braided monoidal categories
two additional structures
- duality
- twist
construction of isotopy invariants of knots, links, tangles, whose components are coloured with objects of a ribbon category

Bakalov-Kirillov p.34
let $\mathfrak{g}$ be a simple Lie algebra
non-trivial example of a ribbon category is provided by the category of finite-dimensional representations of the quantum group $U_q(\mathfrak{g})$
balancing $\delta_V = q^{2\rho} :V \to V^{**}$
on a weight vector $v$ of weight $\lambda$, $q^{2\rho}$ acts as a multiplication by $q^{\langle \langle 2\rho, \lambda \rangle \rangle}$
we see that $V^{**}\equiv V$ as a vector space, but has a different action of $U_q(\mathfrak{g})$, namely

$$ \pi_{V^{**}}(a) = \pi_{V}(\gamma^2(a))), \, a\in U_q(\mathfrak{g}) $$