Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups

Abstract

We study {\em right-invariant (resp., left-invariant) Poisson quasi-Nijenhuis structures} on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called {\em r-qn structures} on the corresponding Lie algebra $\mathfrak{g}$. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all $r$-$qn$ structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between $r$-$qn$ structures and the generalized complex structures on the Lie algebras $\mathfrak{g}$ and also the solutions of modified Yang-Baxter equation on the double of Lie bialgebra $\mathfrak{g}\oplus {\mathfrak{g}}^{\ast }$. The results are applied to some relevant examples.

arXiv admin note: text overlap with arXiv:1708.00209

References in corpus (1)

• Poisson Quasi-Nijenhuis Manifolds