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

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• ID : Q17102717