The Haantjes Tensor and the Nijenhuis Richardson Bracket

Plática dada por Juan Monterde (Universidad de Valencia) en el marco del evento Graded Algebras, Geometry and related Topics para celebrar el 60 aniversario de Adolfo Sánchez Valenzuela el miércoles 16 de noviembre del 2016 en la Universidad Autónoma de Yucatán (UADY)

Abstract:
It is well known that a vector valued differential 1-form, N, on a manifoldis said to be a Nijenhuis tensor if its Nijenhuis torsion vanishes. It is also more or less well known that Nijenhuis torsion can be expressed by means of a graded Lie bracket defined on the space of vector valued differential forms, the Frolicher-Nijenhuis bracket. Nijenhuis tensors appear in diferent topics like complex structures, almost product structures and, mainly, in biHamiltonian systems, this is, in integrable systems. A vector valued differential 1-form, L, is said to be a Haantjes tensor, if another tensor built from it, the Haantjes torsion, vanishes. Although Haantjes tensors were introduced in 1955 as a way to reformulate the notion of integrability, and thus, they are a generalization of Nijenhuis tensors, it was not until recently that they have been used inconnection with biHamiltonian systems. Our objective in this talk is to express the Haantjes torsion in a compact way by means of two graded brackets, the previous one, the Frolicher-Nijenhuis bracket, and a second one the Nijenhuis-Richardson bracket. This second bracket is also defined, in a pure algebraic way, on the space of vector valued differential forms.

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