All Questions
Tagged with braided-tensor-categories hopf-algebras
22
questions
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
13
votes
4
answers
5k
views
What is the universal enveloping algebra?
Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). ...
- 12.1k
11
votes
0
answers
650
views
What is the role of fiber functor in Deligne's theorem on Tannakian categories?
The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...
- 223
10
votes
0
answers
278
views
What's the relation between half-twists, star structures and bar involutions on Hopf algebras?
A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...
- 5,485
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
4
answers
930
views
The dual of a dual in a rigid tensor category
For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{*})^*$, for $*$ denoting dual? If so, what is a good example.
- 953
7
votes
1
answer
385
views
Do dualizable Hopf algebras in braided categories have invertible antipodes?
A classical result of Larson and Sweedler says that a finite dimensional Hopf algebra over a field has invertible antipode. Does this result extend to the setting of Hopf algebras in braided ...
- 7,097
7
votes
1
answer
287
views
Easy example of a non-symmetric braiding of $\operatorname{Rep}(G)$?
What is the smallest group $G$ such that $\operatorname{Rep}(G)$ has a non-symmetric braiding (or just an easy example)?
I seem to remember a result classifying all universal $R$-matrices of $\mathbb ...
- 301
6
votes
2
answers
273
views
When are the braid relations in a quasitriangular Hopf algebra equivalent?
Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations:
$$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$
$$(\mathrm{id} \otimes \Delta) (R) = R_{13} R_{12}...
- 5,485
5
votes
1
answer
323
views
Semisimple Hopf algebras with commutative character ring
Suppose that $A$ is a semisimple Hopf algebra with a commutative character ring. Does it follow that $A$ is quasitriangular, i.e $\mathrm{Rep}(A)$ is a braided tensor category?
I think I 've seen ...
- 837
5
votes
1
answer
439
views
Deligne Tensor Product of Categories, Explicit Equivalence of $A\otimes_\mathbb{C} B\text{-Mod} \cong A\text{-Mod}\boxtimes B\text{-Mod}$
$\newcommand\Mod[1]{#1\text{-Mod}}$Does any one have a reference on a explicit equivalence between
$$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$
The proof in "Tensor Categories ...
- 81
5
votes
1
answer
446
views
Braided monoidal category, example
Let $H$ be a cocommutative hopf algebra.
Let $M$ be the category of $H$-bimodules.
Does the category $M$ form a braided monoidal category with tensor product $\otimes_{H}$ ?
- 69
4
votes
0
answers
99
views
Tensor algebras in the bicategory $\mathsf{2Vect}$
To my knowledge there are two main approaches to categorify the notion of a vector space. I will refer to them as BC-2-vector spaces (Baez, Crans) and KV-2-vector spaces (Kapranov, Voevodsky). Both ...
- 1,527
4
votes
0
answers
101
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 ...
- 815
3
votes
2
answers
311
views
How well is the classification of low-dimensional semisimple Hopf superalgebras (or braided Hopf algebras) understood?
As far as I know, low-dimensional semisimple Hopf algebras are classified (along with non-semisimple ones) up to dimension 60, with the first example of a semisimple Hopf algebra not coming from a ...
- 5,485
3
votes
1
answer
133
views
Integrals and finite dimensionality in braided Hopf algebras
Let $H$ be a Hopf algebra with invertible antipode. Let $A$ be a braided Hopf algebra in the Yetter-Drinfeld category ${}_H^H\mathcal{YD}$ over $H$.
A nonzero left integral in $A$ is a nonzero ...
- 974
2
votes
1
answer
185
views
Comodule Morita equivalence for Hopf algebras
Let $A$ and $B$ be two Hopf algebras, and denote by $\mathcal{M}^A$ and $\mathcal{M}^B$ their respective categories of right comodules. If we have a monoidal equivalence between $\mathcal{M}^A$ and $\...
- 1,104
2
votes
0
answers
60
views
Integrals in noncommutative graded algebras which are not necessarily Hopf
Let $\mathbf{k}$ be a field. Let $A$ be a finite dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbf{k}$-algebra such that $A^0=\mathbf{k}1$. Let $m$ be the maximal non-negative integer such that $A^m\...
- 974
1
vote
0
answers
98
views
Lagrangian subcategories of (non-pointed) braided tensor categories
I am interested in generalising the following claim in On braided fusion categories I (Remarks 4.67.)
“A braided fusion category $\mathcal C$ may have more than one Lagrangian subcategory. E.g., if $\...
- 113
1
vote
0
answers
96
views
When is the action of the braid group on tensor powers of Yetter-Drinfeld modules faithful?
Let $V$ be a Yetter-Drinfeld module over a Hopf algebra $H$ with invertible antipode. Recall that $V$ is a braided vector space with braiding $\Psi\colon V\otimes V\to V\otimes V, v\otimes w\mapsto v_{...
- 974
1
vote
0
answers
84
views
On reflexive bialgebras
Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...
- 974
1
vote
0
answers
123
views
References of an operator $T: V \otimes V \to V \otimes V$
Let $V$ be a vector space with a basis $v_1, \ldots, v_n$ and let $X_{ij} = v_i \otimes v_j$. Then $X_{ij}, i,j=1,\ldots, n$, is a basis of $V \otimes V$. Let $T: V \otimes V \to V \otimes V$ be the ...
- 6,041