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Let $V$ be a vector space with a basis $v_1, \ldots, v_n$ and let $X_{ij} = v_i \otimes v_j$. Then $X_{ij}, i,j=1,\ldots, n$, is a basis of $V \otimes V$. Let $T: V \otimes V \to V \otimes V$ be the linear operator defined by \begin{align} & T(X_{ij}) = q^{-\delta_{ij}}X_{ji}, \\ & T(X_{ji}) = q^{\delta_{ij}} X_{ij} - (q-q^{-1})X_{ji}, \end{align} where $1 \leq i \leq j \leq n$. Then $T$ satisfies $(T-q^{-1})(T+q)=0$ and $T_{12}T_{23}T_{12} = T_{23}T_{12}T_{23}$. The operator $T$ is on page 26 of the paper. Are there some other references about this operator $T$? Thank you very much.

Edit: here $q$ is a non-zero complex number.

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    $\begingroup$ @Steven Gubkin, thank you very much for your comments. I edited the post. $\endgroup$
    – Jianrong Li
    Commented May 18, 2016 at 14:03
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    $\begingroup$ @Steven Gubkin, the first line of the definition of $T$ is for $T(X_{ij})$ and the second line is for $T(X_{ji})$, $i \leq j$. But it is slightly strange since for $i<j$, we have $T(X_{ij})=0$. $\endgroup$
    – Jianrong Li
    Commented May 18, 2016 at 14:05
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    $\begingroup$ For people who haven't followed the link and find the equations dense, this is just the braiding for the standard representation of quantum sl(n) in its usual basis. $\endgroup$
    – Noah Snyder
    Commented May 18, 2016 at 14:34
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    $\begingroup$ I'm not sure what your actual question is. The paper gives several references to a paper of Jimbos, and this is material that would be covered by most textbooks on quantum groups. Have you tried looking at Jimbo's paper? Have you read any textbooks on quantum groups? A keyword that might help you in finding material is "R-matrix". $\endgroup$
    – Noah Snyder
    Commented May 18, 2016 at 14:36
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    $\begingroup$ Note the braiding is R composed with swap. That explains why the middle 2-by-2 is different. Otherwise it's just signs and conventions. $\endgroup$
    – Noah Snyder
    Commented May 19, 2016 at 17:01

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