Generalized Knizhnik-Zamolodchikov equations and the factorization of their solutions
Generalized Knizhnik-Zamolodchikov equations represent the class of Fuchsian systems on \(\mathbb{C}^n\), \(n \geq 2\), that are invariant under the complexification of the action of the Weyl group for a root system in \(\mathbb{R}^n\). The peculiarities of these equations consist in the complexification of the union of hyperplanes orthogonal to the roots of a given root system. Kohno noted that the coefficients of these equations are classical \(r\)-matrices and proposed to construct such equations by the Belavin-Drinfeld method for the theory of \(r\)-matrices. Cherednik established the connection of classical \(r\)-matrices with systems of roots of type \(A_n\), \(n \geq 2\), introduced the concept of a \(r\)-matrix for an arbitrary system of roots, and also deduced an analog of the Knizhnik-Zamolodchikov equation for an arbitrary system of roots. In the present note we show that, as in the case considered by Kohno, the coefficients of simplest generalized Knizhnik-Zamolodchikov equations are generalized \(r\)-matrices. Moreover, we prove the equivalence of the complete integrability conditions for generalized Knizhnik-Zamolodchikov equations and for infinitesimal relations in generalized pure braid groups. Finally, we specify Cherednik's assertion on the local behavior of fundamental solutions of generalized Knizhnik-Zamolodchikov equations and the monodromy of these solutions.
scientific article; zbMATH DE number 795596