Algebraic Nijenhuis operators and Kronecker Poisson pencils

Abstract:
We give a criterion of (micro-)kroneckerity of the linear Poisson pencil  on $g^*$ related to an algebraic Nijenhuis operator $N: g \to g$ on a finite-dimensional Lie algebra $g$. 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 $gl_n$.