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

defn

A ribbon category is a rigid braided tensor category with functorial isomorphisms $\delta_V : V \simeq V^{**}$ satisfying $$ \begin{aligned} \delta_{V\otimes W} & = \delta_V\otimes \delta_W, \\ \delta_{1} & = \operatorname{id}, \\ \delta_{V^{*}} & = (\delta_V^{*})^{-1} \end{aligned} $$ where for $f\in \operatorname{Hom}(U,V)$, $f^*\in \operatorname{Hom}(V^*,U^*)$

example

category of finite-dimensional representations of the quantum group

• 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 $\delta_V = q^{2\rho} :V \simeq V^{**}$
• 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 $V^{**}\simeq V$ as a vector space, but has a different action of $U_q(\mathfrak{g})$, namely

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

• we have $\gamma^2(a) = q^{2\rho}a q^{-2\rho},\, a\in U_q(\mathfrak{g})$

related items

• Modular tensor category