"Ribbon category"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
imported>Pythagoras0 |
||
| 9번째 줄: | 9번째 줄: | ||
==example== | ==example== | ||
* Bakalov-Kirillov p.34 | * 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 $U_q(\mathfrak{g})$ | * 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 $\delta_V = q^{2\rho} :V \to V^{**}$ | * balancing $\delta_V = q^{2\rho} :V \to V^{**}$ | ||
| 17번째 줄: | 18번째 줄: | ||
$$ | $$ | ||
* $\gamma^2(a) = q^{2\rho}a q^{-2\rho}$ | * $\gamma^2(a) = q^{2\rho}a q^{-2\rho}$ | ||
| − | |||
==related items== | ==related items== | ||
2016년 12월 13일 (화) 22:53 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
example
- 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
$$ \pi_{V^{**}}(a) = \pi_{V}(\gamma^2(a))), \, a\in U_q(\mathfrak{g}) $$
- $\gamma^2(a) = q^{2\rho}a q^{-2\rho}$