Questions tagged [r-matrix]

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6
votes
1answer
267 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)...
3
votes
1answer
201 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 ...
11
votes
1answer
600 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 ...
4
votes
0answers
99 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 ...
14
votes
6answers
1k 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 ...
8
votes
4answers
581 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 ...
12
votes
3answers
707 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 ...
3
votes
4answers
863 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 ...
4
votes
2answers
571 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 ...