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Results tagged with integrable-systems
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user 118681
5
votes
1
answer
74
views
Weakly involutive $R$-matrices and representations of the symmetric group $S_N$ in restricte...
An $R$-matrix is a matrix $R\in\operatorname{End}(V\otimes V)$ (where $V$ is a finite dimensional vector space) that solves the Yang–Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
where bo …
- 727
6
votes
2
answers
293
views
Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)
I'm interested in solutions to the Yang-Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the sym …
- 727
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 $m\times m$ m …
- 727