"Ribbon category"의 두 판 사이의 차이

2021년 2월 17일 (수) 01:40 기준 최신판

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

Drinfeld category

related items

Modular tensor category

메타데이터

위키데이터

ID : Q17102717

Spacy 패턴 목록

[{'LOWER': 'ribbon'}, {'LEMMA': 'category'}]
[{'LOWER': 'tortile'}, {'LEMMA': 'category'}]

@@ 6번째 줄: / 6번째 줄: @@
 * 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 <math>\delta_V : V \simeq V^{**}</math> satisfying
+:<math>
+\begin{aligned}
+\delta_{V\otimes W} & = \delta_V\otimes \delta_W, \\
+\delta_{1} & = \operatorname{id}, \\
+\delta_{V^{*}} & = (\delta_V^{*})^{-1}
+\end{aligned}
+</math>
+where for <math>f\in \operatorname{Hom}(U,V)</math>, <math>f^*\in \operatorname{Hom}(V^*,U^*)</math>
 ==example==
+===category of finite-dimensional representations of the quantum group===
 * Bakalov-Kirillov p.34
-* let $\mathfrak{g}$ be a simple Lie algebra
+* 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 $U_q(\mathfrak{g})$
+* 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 $\delta_V = q^{2\rho} :V \simeq V^{**}$
+* balancing <math>\delta_V = q^{2\rho} :V \simeq V^{**}</math>
-* on a weight vector $v$ of weight $\lambda$, $q^{2\rho}$ acts as a multiplication by $q^{\langle \langle 2\rho, \lambda \rangle \rangle}$
+* 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 $V^{**}\simeq V$ as a vector space, but has a different action of $U_q(\mathfrak{g})$, namely
+* 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
-$$
+:<math>
 \pi_{V^{**}}(a) = \pi_{V}(\gamma^2(a))), \, a\in U_q(\mathfrak{g})
-$$
+</math>
-* we have $\gamma^2(a) = q^{2\rho}a q^{-2\rho},\, a\in U_q(\mathfrak{g})$
+* we have <math>\gamma^2(a) = q^{2\rho}a q^{-2\rho},\, a\in U_q(\mathfrak{g})</math>
+===Drinfeld category===
 ==related items==
@@ 23번째 줄: / 35번째 줄: @@
 [[분류:quantum groups]]
+[[분류:migrate]]
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q17102717 Q17102717]
+===Spacy 패턴 목록===
+* [{'LOWER': 'ribbon'}, {'LEMMA': 'category'}]
+* [{'LOWER': 'tortile'}, {'LEMMA': 'category'}]