Integrable Volterra hierarchies over nonabelian algebras

Abstract

Known integrable systems with noncommutative dependent variables are typically formulated over free associative algebras, quantum algebras, or Grassmann algebras. For differential-difference integrable equations, we identify a new class of noncommutative algebras that is compatible with the dynamics and can be positioned between quantum and free algebras. In this brief communication, we consider reductions of the nonabelian Volterra hierarchy to new algebras. This approach extends to a broad class of integrable systems, including the Toda lattice, the Ablowitz-Ladik system, and many others.

This paper is a translation of a Russian manuscript accepted for publication in Russian Mathematical Surveys