Noncommutative Pfaffians associated with the orthogonal algebra

Commutators of Pfaffians associated with the orthogonal algebra are found in skew-symmetric and root realizations of $\mathfrak{o}_N$. A generating function of Pfaffians is proved to satisfy the reflection equation. A relation between Pfaffians in skew-symmetric and root realizations of $\mathfrak{o}_N$ is established. Using these results we construct an integrable equation of Knizhnik-Zamolodchikov type using the Capelli central elements in $U(\mathfrak{o}_N)$, which are sums of squares of the considered Pfaffians. A classical limit of the obtained Knizhnik-Zamolodchikov type equation turns out to be a very specific system of equations of isomonodromic deformations.
Bibliography: 18 titles.