# Symmetries and reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies

## TL;DR

This paper develops the theory of symmetries and hierarchies in noncommutative integrable systems, extending classical KP and Gelfand-Dickey hierarchies to the noncommutative setting with new systems and symmetry structures.

## Contribution

It constructs additional symmetry flows, hierarchies, and new noncommutative systems, including self-consistent sources and string equations, generalizing classical integrable systems.

## Key findings

- Constructed additional flows forming a $W_{1+
ext{infinity}}$ algebra.
- Developed noncommutative KP hierarchy with sources and constraints.
- Derived string equations for noncommutative Gelfand-Dickey hierarchy.

## Abstract

In this paper, we construct the additional flows of the noncommutative Kadomtsev-Petviashvili(KP) hierarchy and the additional symmetry flows constitute an infinite dimensional Lie algebra $W_{1+\infty}$. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions and this generating symmetry is used to construct the noncommutative KP hierarchy with self-consistent sources and the constrained noncommutative KP hierarchy. The above results will be further generalized to the noncommutative Gelfand-Dickey hierarchies which contains many interesting noncommutative integrable systems such as the noncommutative KdV hierarchy and noncommutative Boussinesq hierarchy. Meanwhile, we construct two new noncommutative systems including odd noncommutative C type Gelfand-Dickey and even noncommutative C type Gelfand-Dickey hierarchies. Also using the symmetry, we can construct a new noncommutative Gelfand-Dickey hierarchy with self-consistent sources. Basing on the natural differential Lax operator of the noncommutative Gelfand-Dickey hierarchy, the string equations of the noncommutative Gelfand-Dickey hierarchy are also derived.