All Questions
Tagged with qa.quantum-algebra monoidal-categories
34 questions
31
votes
8
answers
5k
views
Resources for graphical languages / Penrose notation / Feynman diagrams / birdtracks?
There is an idea I've recently gotten interested in that doesn't seem to have a good agreed-upon name ("diagrammatic algebra?"). It centers around the use of two-dimensional diagrams of dots, ...
23
votes
5
answers
3k
views
Do all 3D TQFTs come from Reshetikhin-Turaev?
The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum ...
- 27.5k
18
votes
2
answers
1k
views
Why does Drinfeld Unitarization work?
In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
- 183
16
votes
1
answer
431
views
Are there small examples of non-pivotal finite tensor categories?
I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with.
Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...
- 28.1k
15
votes
6
answers
2k
views
How do I describe a fusion category given a subfactor?
I felt like following up on Kate's question. There were some good motivational answers there.
Given a pair of factors M < N, there is a standard way to construct a 2-category whose objects are M ...
- 45.7k
15
votes
1
answer
729
views
Associators, Grothendieck-Teichmüller group and monoidal categories
The standard definition of an associator seems to be that it a a grouplike power series in two variables $x$ and $ y $ satisfying some pentagon and hexagon relations.
In other words, denoting by $ \...
- 1,033
14
votes
1
answer
880
views
Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories
Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all $...
- 195
12
votes
2
answers
409
views
Reference for free symmetric monoidal categories with duals on symmetric monoidal categories
The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons.
In ...
- 37.8k
9
votes
1
answer
2k
views
Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects
Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.
Questions: are there definitions of image and kernel for a ...
9
votes
2
answers
362
views
Monoidal Equivalence for Drinfeld--Jimbo Quantum Groups
For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra ...
- 219
8
votes
2
answers
852
views
Is a Hopf algebra a group object of some category?
The page of ncatlab on group object states that:
A group object in $\mathrm{CRing}^{\mathrm{op}}$ is a commutative Hopf
algebra.
Question: Is a (noncommutative) Hopf algebra a group object of some ...
8
votes
2
answers
819
views
Are there interesting monoidal structures on representations of quantum affine algebras?
Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...
- 45.7k
7
votes
0
answers
300
views
Does the pentagon axiom force the associativity constraint to be a natural isomorphism?
Consider a fusion ring and the associated system of polynomial equations induced by the pentagon axiom of a fusion category. A solution of this system is supposed to encode the associativity ...
7
votes
0
answers
172
views
When is Rep(U_q(g)) invariant under q -> -q and why?
Since this is a question about quantum groups, let me first fix notation. In this question I use the conventions from this paper of Sawin. (That particular paper conveniently lists in the ...
- 28.1k
5
votes
1
answer
903
views
What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?
For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if
$$(R\otimes id)(id\otimes R)(R\otimes id)
= (id\otimes R)(R\otimes ...
- 1,144
5
votes
5
answers
767
views
What are the correct axioms for a "weakly associative monoidal functor"?
Definitions and the main question
Recall that a category $\mathcal C$ is monoidal if it is equipped with the following data (two functors, three natural transformations, and some properties):
a ...
- 54.6k
4
votes
1
answer
445
views
A submodule of a tensor product of $U_q^{\prime}(\mathfrak{g})$-modules
Does anyone have a proof for the following Lemma?
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and $U_q^{\prime}(\mathfrak{g})$ be the quantum affine algebra over $\...
- 353
4
votes
1
answer
168
views
Link invariants from modular categories (strictification and computation)
By the theory of Reshetikhin and Turaev, a modular tensor category $C$ gives rise to a link invariant. While $C$ is strict as a monoidal category (e.g. $\mathbb{Fib}$), calculating the link can be ...
- 5,230
4
votes
0
answers
320
views
The fusion categories with a strict skeleton
We refer to the book Tensor Categories (by Etingof-Gelaki-Nikshych-Ostrik) for all the notions mentioned in this post.
A fusion category is skeletal if two isomorphic objects are always equal. Every ...
4
votes
0
answers
103
views
Scaling Yetter--Drinfeld Modules
A braided vector space is a pair $(V,\sigma)$ consisting of a vector space $V$, and a linear map $\sigma:V \otimes V \to V \otimes V$, satisfying the Yang--Baxter equation. Ee can scale the braiding ...
- 835
4
votes
0
answers
310
views
Nichols Algebras as Braided Hopf Algebras
Given a Hopf algebra $H$ and a Yetter--Drinfeld module $V$ over $H$, it is well-known that $V$ has an induced braided vector space structure, and so, one can consider it's Nichols algebra which is a ...
- 159
4
votes
0
answers
89
views
Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?
This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.
A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes ...
- 5,565
4
votes
0
answers
134
views
Ribbon Algebras and Co-(dual)-quasi-triangular Hopf Algebras
As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g})}$ the structure of a braided monoidal ...
- 1,453
4
votes
0
answers
300
views
Reshetikhin-Turaev and links with a distinguished component
Hi,
This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question.
Let $T$ be the category whose objects are ...
- 8,524
3
votes
1
answer
218
views
A twisted Haagerup category without pivotal structure
Let $G$ be a finite group, $\tau$ a group automorphism of $G$ of period two and $m$ a natural number. Following [1, Definition 2.1], a complex fusion category $\mathcal{C}$ is called a quadratic ...
3
votes
1
answer
456
views
Rigidity for the category of comodules over a Hopf algebra
On this page
https://ncatlab.org/nlab/show/rigid+monoidal+category
there is a discussion of rigidity (left-right duality) for the catagory of
modules over a Hopf algebra. What happens if we look at ...
- 993
3
votes
0
answers
99
views
Isomorphic objects have the same dimension (pivotal categories)
I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ are isomorphic, then $X$ and $Y$ have the same dimension, i.e.,
$$
\mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \...
- 143
3
votes
0
answers
109
views
Are the fusion categories weakly Frobenius?
A well-known open problem (generalizing Kaplansky 6th conjecture) asks whether every (spherical) fusion category $\mathcal{C}$ (over $\mathbb{C}$) is of Frobenius type, i.e. for every simple object $X$...
2
votes
1
answer
197
views
When are Morita classes represented by certain structured algebra objects?
Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf ...
- 1,209
2
votes
1
answer
276
views
Infinitesimal categories and left duality
I have been reading Kassel's Quantum groups and there is something I can not understand.
In Section 4 of chapter $XX$, he introduces the notion of a Infinitesimal symmetric category, that is
a ...
- 437
2
votes
0
answers
103
views
Questions about proof that all indecomposable module categories over $\operatorname{Rep}(G)$ are equivalent to $\operatorname{Rep}^1(H,\omega)$
$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\End{End}$In Ostrik - Module categories, weak Hopf algebras and modular invariants, it is ...
- 301
2
votes
0
answers
178
views
Categorical dimension and formal codegrees
Let $\mathcal{C}$ be a complex fusion category. If it admits a pivotal structure $a$ then by [1, Proposition 4.7.12], $\dim_a$ induces a character $\chi$ on the Grothendieck ring $Gr(\mathcal{C})$, of ...
1
vote
0
answers
118
views
Are there non-semisimple complex "non-unital special Frobenius algebras"?
I'm interested in "non-unital special Frobenius algebras", consisting of two linear maps (morphisms in the symmetric monoidal category of finite-dimensional complex vector spaces)
$$\mu: V\...
- 3,001
1
vote
0
answers
96
views
Frobenius Monoids as Collapsed 2-Categories
Let $\mathbf{COB}_2$ denote the 2-category given by
$\bullet$ objects are finite sets of points
$\bullet$ 1-morphisms between these are 1d cobordisms
$\bullet$ 2-morphisms are 2d cobordisms with ...
- 433