Haantjes Algebras and Diagonalization

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.

29 pages, no figures

References in corpus (4)

• Nijenhuis Geometry
• Recursion Operators and Frobenius Manifolds
• Haantjes Algebras of the Lagrange Top
• On the decoupling problem of general quasilinear first order systems in two independent variables

Cited by in corpus (8)

• On integrable systems outside Nijenhuis and Haantjes geometry
• Partial separability and symplectic-Haantjes manifolds
• Polynomial Structures in Generalized Geometry
• On the Jordan-Chevalley decomposition problem for operator fields in small dimensions and Tempesta-Tondo conjecture
• Higher Haantjes Brackets and Integrability
• Nijenhuis geometry of parallel tensors
• Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry
• Integrable sigma models with Haantjes structure on $H_4$ Lie group