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The quantum Yang-Baxter equation is $R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$. The braid equation is $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$. It is said that these two equations are equivalent. How to prove this? Thank you very much.

LJR
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1 Answers1

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Let $V$ be a vector space and let $\tau:V\otimes V\to V\otimes V$ to be the flip transformation defined by $v\otimes w\mapsto w\otimes v$ for all $v,w\in V$.

I believe what is meant by saying that the quantum Yang-Baxter and braid equations are "equivalent" is that if $R:V\otimes V\to V\otimes V$ is an invertible linear transformation, then $R$ satisfies one of them if and only if $\tau R$ satisfies the other. In other words, once you have a solution to one, you automatically get a solution to the other using the flip.

Casteels
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  • I don't see how $R$ satisfies one of them if and only if $\tau R$ satisfies the other. Do you have a hint? – Max Demirdilek Aug 24 '20 at 10:40
  • @MaxDemirdilek If ${e_i}i$ is a basis of $V$ and $R(e_i\otimes e_j)=\sum{p,q}R_{ij}^{pq}e_p\otimes e_q$, evaluating the two equations at the points $e_i\otimes e_j\otimes e_k$ yields the same set of equations for the coefficients $R_{ij}^{pq}$. – Noiril Apr 08 '25 at 17:57