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Abstract
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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 ⊕ g∗. The
results are applied to some relevant examples.
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