Haantjes Algebras and Diagonalization

TL;DR

This paper introduces Haantjes algebras, a geometric framework that generalizes known structures and provides conditions for the simultaneous diagonalization or block-diagonalization of operator families on manifolds.

Contribution

It defines Haantjes algebras and demonstrates their role in operator diagonalization, extending geometric structures and unifying diagonalization conditions.

Findings

Existence of local coordinates for simultaneous diagonalization of commuting, semisimple operators.

Block-diagonal form achievable for non-semisimple, commuting operators.

Haantjes algebras generalize structures in Riemannian geometry and integrable systems.

Abstract

We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They are also required to satisfy suitable compatibility conditions. Haantjes algebras naturally generalize several known interesting geometric structures, arising in Riemannian geometry and in the theory of integrable systems. At the same time, as we will show, they play a crucial role in the theory of diagonalization of operators on differentiable manifolds. Assuming that the operators of a Haantjes algebra are semisimple and commute, we shall prove that there exists a set of local coordinates where all operators can be diagonalized simultaneously. Moreover, in the general, non-semisimple case, they acquire simultaneously, in a suitable local chart, a block-diagonal form.