Let \(\mathfrak g\) be a finite dimensional Lie algebra over \(\mathbb{C}\), \(\{x_ a\}\) be an orthonormal basis with respect to the Killing form, \(r(\lambda)\) be a classical nondegenerate \(r\)-matrix. The latter means that \(r(\lambda)=c_ 2\cdot\lambda^{-1}+\sum_ a\rho_ a\otimes x_ a+O(\lambda)\) in a neighborhood of \(\lambda=0\) for \(c_ 2=\sum_ ax_ a\otimes x_ a\), some \(\{\rho_ a\}\subset{\mathfrak g}\) and \([r^{12},r^{13}]+[r^{12},r^{23}]+[r^{13},r^{23}]=0\). Let \(\varphi_{ij}\) be the standard embedding of \({\mathfrak g}^{\otimes 2}\) into \({\mathfrak g}^{\otimes n}\) acting on the \((i,j)\)-place. Let \(R_ i=\sum_{j\neq i}\varphi_{ij}(r(\lambda_ i-\lambda_ j))\in{\mathfrak g}^{\otimes n}\), \(V=\otimes V_ i\) be a \({\mathfrak g}^{\otimes n}\)- module. Then one can define the Knizhnik-Zamolodchikov equation for the given \(r\), namely \(\chi\cdot\partial W/\partial\lambda_ i=R_ i\cdot W\) with \(W\) taking values in \(V\). The fact that \(r\) satisfies the CYBE implies that this system is consistent.
The author considers the Knizhnik-Zamolodchikov equation for \[ r=c_ 2\cdot ct_ u(\lambda)+\sum_ \alpha(e_{-\alpha}\otimes e_ \alpha- e_ \alpha\otimes e_{-\alpha})/(e_ \alpha,e_{-\alpha}), \] where \(e_ \alpha\) is a generator of the root subspace \(E_ \alpha\subset{\mathfrak g}\) and \(ct_ u(\lambda)=u\cdot(e^{u\lambda}+e^{-u\lambda})/(e^{u\lambda}-e^{- u\lambda})\) . Now let \({\mathfrak G}_ i={\mathfrak g}((\lambda_ i))\), \({\mathfrak G}=\prod^ n_{i=1}{\mathfrak G}_ i\) and \(\hat{\mathfrak G}={\mathfrak G}\oplus\mathbb{C}\cdot c\). One can define the Lie algebra structure on \(\hat{\mathfrak G}\). Then \(V\) generates the Verma module \(M\) over \(\hat{\mathfrak G}\). There exists a map \(\pi: M\to V\) constructed in [the author, Funct. Anal. Appl. 19, 193–206 (1985), transl. from Funkts. Anal. Prilozh. 19, No. 3, 36–52 (1985; Zbl 0577.17011)]. Using \(\pi\) the author defines some special function \(w\) with values in \(V\) and the action of the operator \((\chi\cdot\partial/\partial\lambda_ i-R_ i)\) on it. Though \((\chi\cdot\partial/\partial\lambda_ i-R_ i)\cdot w=T\neq 0\), there exists a suitable map \(\operatorname{int}\) such that \(\operatorname{int}(T)=0\). Therefore \(\operatorname{int}(w)\) satisfies the Knizhnik-Zamolodchikov equation.