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Abstract
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We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra . We show that - structures can be used to find compatible solutions of the classical Yang-Baxter equation. Conversely, two compatible r-matrices from which one is invertible determine an - structure. We classify, up to a natural equivalence, all -matrices and all - structures with invertible on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by -matrices on a phase space whose symmetry group is Lie group , can be specifically determined.
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