Questions tagged [r-matrix]
The r-matrix tag has no usage guidance.
12
questions
15
votes
6
answers
2k
views
What structure on a monoidal category would make its 2-category of module categories monoidal and braided?
So, many of us know the answer to "what kind of structure on an algebra would make its category of representations braided monoidal": your algebra should be a quasi-triangular Hopf algebra (maybe if ...
- 43.4k
14
votes
3
answers
824
views
Are supervector spaces the representations of a Hopf algebra?
Supervector spaces look a lot like the category of representations of $\mathbb{Z}/2\mathbb{Z}$ - the even part corresponds to the copies of the trivial representation and the odd part corresponds to ...
- 114k
11
votes
1
answer
798
views
R-matrices, crystal bases, and the limit as q -> 1
I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of ...
- 8,369
8
votes
4
answers
673
views
What's the right object to categorify a braided tensor category?
The yoga of categorification has gained a lot of popularity in recent years, and some techniques for it have made a lot of progress. It's well-understood that, for example, a ring is probably ...
- 43.4k
8
votes
1
answer
279
views
R-matrices and symmetric fusion categories
Given a $\mathbb{C}$-linear braided fusion category $\mathcal{C}$ containing a fusion rule of the form e.g.
\begin{equation}X\otimes Y\cong A\oplus B \oplus C\end{equation}
(where $A,B, C, X$ and $Y$ ...
- 277
6
votes
1
answer
567
views
Classification of quasitriangular Hopf algebras
The classification of hopf algebras is a big and open problem, containing various subproblems (such as: the classification of groups, of Lie algebras, the study of special classes such as (co)...
- 7,628
5
votes
4
answers
1k
views
An inner product that makes the R-matrix unitary
So, if you talk to the right people, they will tell you that the braiding of the category of representations of a quantum group are not unitary and that one can fix this by taking a different commutor ...
- 43.4k
5
votes
2
answers
588
views
How does one think about the "off-diagonal" part of the $R$-matrix?
The universal $R$-matrix of a quantized universal enveloping algebra is typically written as the product of two terms, one only involving elements of the Cartan, and one only involving elements of the ...
- 43.4k
5
votes
0
answers
98
views
How to construct the quantum group $U_q(\mathfrak{sl}(2))$ from the quantum coordinate ring $\operatorname{SL}_q(2)$?
Let $k$ be the ground field, and $q$ be an invertible element with $q$ not being a root of unity.
Let $\operatorname{SL}_q(2)$ be the quantum coordinate ring of $\operatorname{SL}(2)$ given explicitly ...
- 163
4
votes
0
answers
105
views
Is there a good reference for how ribbon structures change when one switches coproducts?
I'm just going assume readers are familiar with the notions of R-matrix and ribbon categories.
Given a quasi-triangular Hopf algebra $A$ with $R$-matrix $R$, one can construct the co-opposite Hopf ...
- 43.4k
3
votes
1
answer
301
views
Yang-Baxter equation for the asymmetric simple exclusion process (ASEP)
On page 14 of Craig Tracy's slides on ASEP, it states that the $n$-particle boundary condition can be reduced to the 2-particle boundary condition due to the fact that the $S$-matrix satisfies the ...
- 247
0
votes
0
answers
27
views
Question regarding orthogonality of a map
I am reading the materials discussed in lecture $5$ from the lecture notes on quantum groups about Belavin-Drinfeld classification theorem written by Pavel Etingof and Oliver Schiffmann.
In the first ...
- 179