Invariant Poisson–Nijenhuis structures on Lie groups and classification
International Journal of Geometric Methods in Modern Physics (IJGMMP)
We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group G and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra g. We show that r-n 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 r-n structure. We classify, up to a natural equivalence, all r-matrices and all r-n structures with
... tible r on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by r-matrices on a phase space whose symmetry group is Lie group G, can be specifically determined.