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 ($t$_{i}t_{j}=q_{ij}^{-1}t_{j}t_{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 $$**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.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k01/k013/k01305h.htm

Last modified: December 23, 2001