Title: Haantjes Algebras and Diagonalization,
Abstract: We propose the notion of Haantjes algebra, which consists of an assignment of
a family of fields of operators over a differentiable manifold, with vanishing
Haantjes torsion and satisfying suitable compatibility conditions among each
others. Haantjes algebras naturally generalize several known interesting
geometric structures, arising in Riemannian geometry and in the theory of
integrable systems. At the same time, they play a crucial role in the theory of
diagonalization of operators on differentiable manifolds.
Whenever the elements of an 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 non-semisimple case, they
acquire simultaneously a block-diagonal form.