Poisson quasi-Nijenhuis manifolds and Dirac structures: A geometrical approach to deformation and involutive theorems - Luiz, Murilo do Nascimento (2024)
In this work, we analyze the connection between Poisson quasi-Nijenhuis structures, quasi-Lie bialgebroids, and Courant algebroids. We demonstrate how to deform a Poisson quasi-Nijenhuis manifold using a closed 2-form within the context of Courant algebroids and Dirac structures. Then, we interpret this procedure in the context of supermanifolds, as a specific instance of the so-called twisting of a proto-bialgebroid. Finally, we investigate the applications of Poisson quasi-Nijenhuis manifolds in the theory of integrable systems. The main results of this thesis are reported in (LUIZ; MENCATTINI; PEDRONI, 2024).