Publication: Higher Haantjes brackets and integrability
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2021-11
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Springer
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Abstract
We propose a new, infinite class of brackets generalizing the Frolicher-Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We also prove that the vanishing of a higher-level Nijenhuis torsion of an operator field is a sufficient condition for the integrability of its eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form.
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CRUE-CSIC (Acuerdos Transformativos 2021)
© The Author(s) 2021
the CRUE-CSIC agreement with Springer Nature.
The authors gratefully thank Prof. Y. Kosmann-Schwarzbach for useful discussions. Also, P. T. wishes to thank heartily Prof. N. Kamran for a careful reading of the manuscript, discussions and encouragement. We also wish to thank the Referees for many helpful suggestions.The research of P. T. has been supported by the research project PGC2018-094898-B-I00, Ministerio de Ciencia, Innovación y Universidades and Agencia Estatal de Investigación, Spain, and by the Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S), Ministerio de Ciencia, Innovación y Universidades y Agencia Estatal de Investigación, Spain. The research of G. T. has been supported by the research project FRA2020-2021, Universitá degli Studi di Trieste, Italy. P. T. is member of the Gruppo Nazionale di Fisica Matematica (GNFM).