An introduction to associative geometry with applications to integrable systems

TL;DR

This paper introduces associative geometry, a differential calculus framework on associative algebras, which offers new insights into classical solution methods for finite-dimensional integrable systems.

Contribution

It presents a novel formalism called associative geometry that enhances understanding of integrable systems through associative algebra calculus.

Findings

Associative geometry provides a new perspective on integrable systems.

The formalism simplifies classical solution methods.

It bridges differential calculus and associative algebra theory.

Abstract

The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory associative geometry. We argue that this formalism sheds a new light on some classic solution methods in the theory of finite-dimensional integrable dynamical systems.