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Invariant Poisson–Nijenhuis structures on Lie groups and classification

Zohreh Ravanpak

Department of Mathematics, Faculty of Science, Azarbaijan Shahid Madani University, Tabriz, Iran

Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

E-mail Address: z.ravanpak@azaruniv.ac.ir

E-mail Address: zravanpak@impan.pl

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Adel Rezaei-Aghdam

Department of Physics, Faculty of Science, Azarbaijan Shahid Madani University, Tabriz, Iran

E-mail Address: rezaei-a@azaruniv.ac.ir

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Ghorbanali Haghighatdoost

Department of Mathematics, Faculty of Science, Azarbaijan Shahid Madani University, Tabriz, Iran

E-mail Address: gorbanali@azaruniv.ac.ir

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https://doi.org/10.1142/S0219887818500597Cited by:9 (Source: Crossref)

Abstract

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ( $P$ - $N$ ) 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 (CYBE). 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 a $G$ , can be specifically determined.

Keywords:

AMSC: 37K05, 37K10, 53D17, 37K30

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