Invariant Poisson-Nijenhuis structures on Lie groups and classification

Abstract

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 $\mathfrak{g}$. 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 invertible $r$ 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.

arXiv comment23 pages, Revised version

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Cited by in corpus (7)

• Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation
• $\mathcal{O}$-operators and related structures on Leibniz algebras
• Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O}N$-structures
• Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs
• Integrable sigma models with complex and generalized complex structures
• Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups
• Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures