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

## TL;DR

This paper explores invariant Poisson quasi-Nijenhuis structures on Lie groups and their infinitesimal counterparts on Lie algebras, classifying these structures on specific four-dimensional Lie algebras and examining their relations to generalized complex structures and Yang-Baxter solutions.

## Contribution

It introduces the concept of r-qn structures on Lie algebras and classifies them for certain four-dimensional cases, linking them to broader geometric and algebraic frameworks.

## Key findings

- Classification of r-qn structures on selected Lie algebras
- Identification of relations with generalized complex structures
- Connections to solutions of the modified Yang-Baxter equation

## 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 $\mathfrak g\oplus\mathfrak g^*$. The results are applied to some relevant examples.