Questions tagged [braided-tensor-categories]
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15
questions
27
votes
3
answers
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views
Quantum group as (relative) Drinfeld double?
The most elementary construction I know of quantum groups associated to a finite dimensional simple Hopf algebra is to construct an algebra with generators $E_i$ and $F_i$ corresponding to the simple ...
- 6,043
10
votes
2
answers
701
views
Is there a non-degenerate quadratic form on every finite abelian group?
Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, ...
- 25.5k
8
votes
2
answers
775
views
Non-symmetric Braiding on finite group Representation Categories
Do the fusion categories $Rep(S_4)$ and $Rep(A_5)$ admit non-symmetric braidings? All the other rep. cats. of finite subgroups of $SU(2)$ do (in the McKay correspondence). My guess is no.
- 1,629
5
votes
2
answers
247
views
What is the proof of the compatibility of a braiding with the unitors?
I am specifically referencing the property that, given a braided monoidal category with a braiding $c$ and left and right unitors $\lambda, \rho$,
$$
\lambda_A \circ c_{A,I}=\rho_{A},
$$
for any ...
- 303
5
votes
1
answer
289
views
Is every premodular category the *full* subcategory of a modular category?
In Müger's article "Conformal Field Theory and Doplicher-Roberts Reconstruction", he defines the "modular closure" of a braided monoidal category. So every braided monoidal category (and therefore ...
- 5,485
22
votes
3
answers
3k
views
How many definitions are there of the Jones polynomial?
Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
- 18.1k
16
votes
4
answers
1k
views
Braided Hopf algebras and Quantum Field Theories
It is well-known, that there are a lot of applications of classical Hopf algebras in QFT, e.g. Connes-Kreimer renormalization, Birkhoff decomposition, Zimmermann formula, properties of Rota-Baxter ...
- 787
12
votes
1
answer
509
views
Is there a "killing" lemma for G-crossed braided fusion categories?
Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different.
Premodular categories
In braided ...
- 5,485
11
votes
1
answer
694
views
Is the category $\operatorname{sVect}$ an "algebraic closure" of $\operatorname{Vect}$?
$\DeclareMathOperator\sVect{sVect}\DeclareMathOperator\Vect{Vect}$The category $\sVect_k$ of (let's say finite-dimensional) super vector spaces can be obtained from the category $\Vect_k$ of (finite-...
- 59k
10
votes
4
answers
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180˚ vs 360˚ Twists in String Diagrams for Ribbon Categories
Ribbon categories are braided monoidal categories with a twist or balance, $\theta_B:B\to B$, which is a natural transformation from the identity functor to itself. In the string diagram calculus for ...
- 1,852
9
votes
4
answers
2k
views
The tensor product of two monoidal categories
Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way?
The motivation I am thinking of is two categories that are representation ...
- 815
9
votes
1
answer
226
views
Cyclic structure on a balanced (or ribbon) monoidal category
As it is well known, a balanced (and in particular ribbon) monoidal category is an algebra over the framed little 2-discs operad. The latter is homotopy equivalent to the operad of moduli space of ...
- 7,962
7
votes
2
answers
875
views
Enrichments vs Internal homs
Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor
$$
X \otimes -: \cal{C} \to \cal{C},
$$
for ...
- 953
4
votes
1
answer
427
views
About a categorical definition of graded (coloured) algebra
The definition of graded algebra had a growing interest in algebra and mathematical physics (see $[GTC]$), I see that this topic has an elegant and simple categorical generalization, but I have not ...
- 4,095
3
votes
0
answers
281
views
About symmetry, braids, and pseudo-functors.
Let $(\mathcal{C}, \otimes, I)$ a monoidal category, and let $\mathbb{B}(\mathcal{C})$ the bicategory (with only one object and $(\mathcal{C}, \otimes, I)$ as (monoidal) category of morphisms and ...
- 4,095