$W$-algebras and higher analogues of the Knizhnik–Zamolodchikov equations

The key role in the derivation of the Knizhnik–Zamolodchikov equations in the Wess–Zumino–Witten model is played by the energy–momentum tensor, which is constructed from a second-order Casimir element in the universal enveloping algebra of the corresponding Lie algebra. We investigate the possibility of constructing analogues of Knizhnik–Zamolodchikov equations using higher-order central elements. We consider the Casimir element of the third order for the Lie algebra $\mathfrak{sl}_N$ and of the fourth order for $\mathfrak{o}_N$. The construction is impossible in the first case, but we succeed in obtaining the sought equation in the second case.