Algebraic Nijenhuis operators and Kronecker Poisson pencils. (English) Zbl 1114.53067
Summary: We give a criterion of (micro-)Kroneckerity of the linear Poisson pencil on \(\mathfrak {g}^{\ast}\) related to an algebraic Nijenhuis operator \(N: \mathfrak {g} \to \mathfrak {g}\) on a finite-dimensional Lie algebra \(\mathfrak {g}\). As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related to Borel subalgebras and a new proof of the complete integrability of the free rigid body system on \(\mathfrak {gl}_n\).
MSC:
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 17B63 | Poisson algebras |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
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