Schur pairs, non-commutative deformation of the Kadomtsev--Petviashvili hierarchy and skew differential operators

E. E. Demidov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 3, pp. 683-698.

Résumé

The concept of Schur pairs emerges naturally when the KP-hierarchy is treated geometrically as a dynamical system on an infinite-dimensional Grassmann manifold. On the other hand, these pairs classify the commutative subalgebras of differential operators. Analyzing these interrelations one can obtain a solution of the classical Schottky problem or a version of the Burchnall–Chaundy–Krichever correspondence. The article is devoted to a non-commutative analogue of the Schur pairs. The author has introduced the KP-hierarchy with non-commutative time space ($t_it_j=q_{ij}^{-1}t_jt_i$) and a non-commutative Grassmann manifold, which form a non-commutative formal dynamical system. The Schur pair $(A,F)$ consists of a subalgebra $A$ of pseudodifferential operators with non-commutative coefficients and a point $F$ of $\mathbf G$ such that $A$ stabilizes $F$. We obtain a transformation law for Schur pairs under non-commutative KP flows. A way of constructing differential operators from a given Schur pair is presented. The commutative subalgebras of differential operators of a special type are classified in terms of Schur pairs.