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

We study right-invariant (respectively, left-invariant) Poisson quasi-Nijenhuis structures on a Lie group G and introduce their infinitesimal counterpart, the so-called r-qn structures on the corresponding Lie algebra 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 g and also the solutions of modified Yang-Baxter equation (MYBE) on the double of Lie bialgebra g circle plus g*. The results are applied to some relevant examples.