Knizhnik-Zamolodchikov equation

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Problem 4.51 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 4 "Applications"

Consider the system of linear partial differential equations (Knizhnik-Zamolodchikov equation)

\frac{\partial }{\partial z_i} \Phi = k \sum _{j \neq i}^{n} \frac{P^{(ij)}}{z_i-z_j} \Phi

where $\Phi = \Phi (z_1,\ldots ,z_n)$ is a function of $n$ complex independent variables, which takes values in the tensor product $V \otimes V \otimes \cdots \otimes V$ of $N$ identical vector spaces $V$ . $P^{(ij)}$ is the permutation of the $i$ -th and $j$ -th factors of the tensor product. This is a simple form of the KZ-equation. Show that the KZ-equation is consistent. Consider the differential operators

O_i := \frac{\partial }{\partial z_i} - k \sum _{j \neq i}^{n} \frac{P^{(ij)}}{z_i-z_j}, \quad i = 1,\ldots ,n .

Show that $O_i$ and $O_j$ commute.

References

[STH12] Steeb W., Tanski I., Hardy Y. Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications. World Scientific Publishing, 2012.