FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2001, VOLUME 7, NUMBER 3, PAGES 683-698

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

E. E. Demidov

Abstract

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```
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.
```

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Location: http://mech.math.msu.su/~fpm/eng/k01/k013/k01305t.htm.

Last modified: December 23, 2001