All Questions
Tagged with braided-tensor-categories rt.representation-theory
17
questions
18
votes
2
answers
4k
views
What is a tensor category?
A monoidal category is a well-defined categorical object abstracting products to the categorical setting. The term tensor category is also used, and seems to mean a monoidal category with more ...
- 815
11
votes
1
answer
336
views
What is the relation between 2-Gerstenhaber, CohFT, and Gerstenhaber geometrically?
Background. As we know from Fred Cohen's Thesis, taking homology of the little 2-discs operad $\mathcal{D}_2$ with coefficients in a field of characteristic zero produces the Gerstenhaber operad $\...
- 1,971
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
2
answers
1k
views
Drinfeld't map, centre of quantum group, representation category of quantum group
My question is about the Drinfeld't map between $Rep(U_q(\mathfrak{g}))$ and $Z(U_q(\mathfrak{g}))$. I have heard the reference 1989 paper by Drinfeld't "Almost cocommutative Hopf algebras" - but this ...
- 2,136
7
votes
2
answers
195
views
Where does the univeral $R$-matrix of $U_q(\mathfrak g)$ live?
Let $\mathfrak g$ be a complex simple Lie algebra and let $U_q(\mathfrak g)$ denote the Drinfeld-Jimbo quantum group associated to $\mathfrak g$. I will assume that $U_q(\mathfrak g)$ is a $\mathbb C(...
- 691
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
244
views
What is the explanation for the special form of representations of three string braid group constructed using quantum groups information supplied
It is well-known that representations of quantised enveloping algebras give representations of braid groups. For the examples that I know explicitly the representations of the three string braid group ...
- 9,032
6
votes
0
answers
126
views
Braid groups representations on infinite dimensional vector spaces
Let $V$ be an infinite dimensional complex vector space. Let $R:V\otimes V\to V\otimes V$ be a solution to the quantum Yang Baxter Equation. In other words: $R$ is invertible and satisfies the ...
- 4,969
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
2
answers
507
views
Representation theory in braided monoidal categories
The crux of what I wish to know is what results from representation theory, a subject usually framed within the category $\text{Vect}_\mathbb{k}$, follow in more general braided monoidal categories? I ...
- 191
4
votes
0
answers
113
views
semisimplicity of maps in braided vector spaces
Let $V$ be a finite dimensional braided vector space over $\mathbb{C}$.
This means that we have a map $$c_{V,V}:V\otimes V\to V\otimes V$$ which gives us an action of the braid group $B_n$ on $V^{\...
- 4,969
3
votes
1
answer
240
views
representation of a group and its center
(I asked the following question at StackExchange but received no answer.)
Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional ...
user46846
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
2
votes
1
answer
147
views
Modularisation on group representations with arbitrary braiding
Applying the modularisation/deequivariantisation procedure to the representation category $\operatorname{Rep}_G$ of a finite group $G$ with trivial braiding gives the fibre functor to vector spaces. ...
- 5,485
1
vote
1
answer
118
views
Braided R-matrices for finite action groupoids
1. Algebra from action groupoids
Let $G$ be a finite group acting on a finite set $X$ from the
right (denoted in element as $x^{g}$). We have an algebra (of the
action groupoid) over $\mathbb{C}$: the ...
- 4,760
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
0
votes
0
answers
65
views
Is a Lagrangian subgroup of a metric group isomorphic to its quotient?
A metric group is a finite abelian group $G$ with a quadratic function
$$q:G\rightarrow \mathbb R/\mathbb Z\;,$$
that is,
$$M(a,b):= q(a+b)-q(a)-q(b)$$
is bilinear in $a$ and $b$ [edit: and non-...
- 2,839