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Questions tagged [r-matrix]

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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 ...
matha's user avatar
  • 183
9 votes
1 answer

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 $Y$ ...
Meths's user avatar
  • 309
6 votes
1 answer

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)...
Konstantinos Kanakoglou's user avatar
3 votes
1 answer

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 ...
Y. Pei's user avatar
  • 247
12 votes
1 answer

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 ...
MTS's user avatar
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4 votes
0 answers

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 ...
Ben Webster's user avatar
  • 44.2k
15 votes
6 answers

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 ...
Ben Webster's user avatar
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8 votes
4 answers

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 ...
Ben Webster's user avatar
  • 44.2k
15 votes
3 answers

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 ...
Qiaochu Yuan's user avatar
5 votes
4 answers

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 ...
Ben Webster's user avatar
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5 votes
2 answers

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 ...
Ben Webster's user avatar
  • 44.2k