"Ribbon category"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 7번째 줄: | 7번째 줄: | ||
;defn | ;defn | ||
| − | A ribbon category is a rigid braided tensor category with functorial isomorphisms | + | A ribbon category is a rigid braided tensor category with functorial isomorphisms <math>\delta_V : V \simeq V^{**}</math> satisfying |
| − | + | :<math> | |
\begin{aligned} | \begin{aligned} | ||
\delta_{V\otimes W} & = \delta_V\otimes \delta_W, \\ | \delta_{V\otimes W} & = \delta_V\otimes \delta_W, \\ | ||
| 14번째 줄: | 14번째 줄: | ||
\delta_{V^{*}} & = (\delta_V^{*})^{-1} | \delta_{V^{*}} & = (\delta_V^{*})^{-1} | ||
\end{aligned} | \end{aligned} | ||
| − | + | </math> | |
| − | where for | + | where for <math>f\in \operatorname{Hom}(U,V)</math>, <math>f^*\in \operatorname{Hom}(V^*,U^*)</math> |
==example== | ==example== | ||
===category of finite-dimensional representations of the quantum group=== | ===category of finite-dimensional representations of the quantum group=== | ||
* Bakalov-Kirillov p.34 | * Bakalov-Kirillov p.34 | ||
| − | * let | + | * let <math>\mathfrak{g}</math> 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 | + | * non-trivial example of a ribbon category is provided by the category of finite-dimensional representations of the quantum group <math>U_q(\mathfrak{g})</math> |
| − | * balancing | + | * balancing <math>\delta_V = q^{2\rho} :V \simeq V^{**}</math> |
| − | * on a weight vector | + | * on a weight vector <math>v</math> of weight <math>\lambda</math>, <math>q^{2\rho}</math> acts as a multiplication by <math>q^{\langle \langle 2\rho, \lambda \rangle \rangle}</math> |
| − | * we see that | + | * we see that <math>V^{**}\simeq V</math> as a vector space, but has a different action of <math>U_q(\mathfrak{g})</math>, namely |
| − | + | :<math> | |
\pi_{V^{**}}(a) = \pi_{V}(\gamma^2(a))), \, a\in U_q(\mathfrak{g}) | \pi_{V^{**}}(a) = \pi_{V}(\gamma^2(a))), \, a\in U_q(\mathfrak{g}) | ||
| − | + | </math> | |
| − | * we have | + | * we have <math>\gamma^2(a) = q^{2\rho}a q^{-2\rho},\, a\in U_q(\mathfrak{g})</math> |
===Drinfeld category=== | ===Drinfeld category=== | ||
2020년 11월 13일 (금) 06:59 판
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 \(\gamma^2(a) = q^{2\rho}a q^{-2\rho},\, a\in U_q(\mathfrak{g})\)