13
votes
Accepted
Why does the definition of a braided monoidal category not mention the braid equation?
Indeed, the axioms of a braided monoidal category are enough to derive the Yang-Baxter equation. See Braided monoidal categories by Joyal and Street (diagram B7), or 1Lab for a formalised proof.
- 468
11
votes
Accepted
What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?
Your two equations are equivalent, and are both versions of the quantum YBE. (The question from the comments does a good job of answering your classical versus quantum question.)
Write the first as
$$
...
- 2,533
3
votes
Weakly involutive $R$-matrices and representations of the symmetric group $S_N$ in restricted subspaces of $V^{\otimes N}$
In the recent physics preprint
Corcoran, De Leeuw and Pozsgay, Integrable models on Rydberg atom chains [arXiv:2405.15848]
the authors study quantum-integrable models related to $R$-matrices with ...
- 1,996
3
votes
Accepted
Set-theoretic solutions of YBE for $n=3$
Sure,
degenerate solutions:
\begin{equation}
((1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)),
((1, 1), (1, 2), (3, 1), (2, 1), (2, 2), (3, 2), (1, 3), (2, 3), (3, 3)),
((1, 1),...
- 46
2
votes
Yang-Baxter equation for the asymmetric simple exclusion process (ASEP)
The $S$-matrix you write can be written as $\frac{x_\alpha-Q x_\beta}{x_\alpha-x_\beta}$, where $x_{\alpha,\beta}^{}$ are some fractional linear transformations of $\xi_{\alpha,\beta}^{}$, and $Q$ is ...
- 1,765
2
votes
Accepted
How can I verify that a given solution of the Quantum Yang-Baxter equation is associated to a given Lie algebra?
To the best of my knowledge this is a very hard problem and the answer to this question is, unfortunately, open.
A famous example that illustrates this in the context of quantum integrability comes ...
- 1,996
2
votes
Accepted
Solution of the Yang-Baxter equation associated to the $U_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra
Drinfel'd's quantum double is a construction that produces, given a Hopf algebra, an $R$-matrix that turns it into a quasi-triangular Hopf algebra. You could try working that out to get an $R$-matrix ...
- 1,996
1
vote
Accepted
Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)
Here is a family of examples indexed by an integer $m\geq 3$.
Let
\begin{eqnarray}
R^{ij}_{kl}=\lambda_{ij}c_{kl}-\delta_{ik}\delta_{jl},\tag{1}
\end{eqnarray}
where $\lambda, c$ are $m\times m$ ...
- 727
1
vote
Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)
I am not sure about the conjugation property that you are asking, but I know some solutions to the set theoretical YB equation, which are degenerate, nevertheless interesting. The simplest one comes ...
- 293
1
vote
Examples of Yang-Baxter monoids
What about using the theory of braces? A skew brace is a triple $(A,+,\circ)$, where $(A,+)$ and $(A,\circ)$ are groups and $a\circ (b+c)=a\circ b-a+a\circ c$ holds for all $a,b,c\in A$. Notation: If $...
- 3,078
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