Algebraic Nijenhuis operators and Kronecker Poisson pencils

We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on $\frak{g}^*$ related to an algebraic Nijenhuis operator $N:\frak{g}\to \frak{g}$ on a finite-dimensional Lie algebra $\frak{g}$. 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 $\frak{gl}_n$.Comment: 10 pages, references adde

Veronese webs for bihamiltonian structures of higher corank

It is shown how the well-known class of bihamiltonian structures in general position can be extended to a wider class. A generalization of the corresponding notion of a Veronese web for this wider class is presented (in the general position case Veronese webs form complete systems of local invariants for bihamiltonian structures). Some examples are considered.Comment: 11p. To appear in: Banach Center Publications, Proc. of "Poisson Geometry" conference dedicated to the memory of Stanislaw Zakrzewski, Warsaw 1998, J.Grabowski, P.Urbanski ed

On integrability of generalized Veronese curves of distributions

Given a 1-parameter family of 1-forms \g(t)= \g_0+t\g_1+...+t^n\g_n, consider the condition d\g(t)\wedge\g(t)=0 (of integrability for the annihilated by \g(t) distribution $w(t)$). We prove that in order that this condition is satisfied for any $t$ it is sufficient that it is satisfied for $N=n+3$ different values of $t$ (the corresponding implication for $N=2n+1$ is obvious). In fact we give a stronger result dealing with distributions of higher codimension. This result is related to the so-called Veronese webs and can be applied in the theory of bihamiltonian structures.Comment: 7p., to appear in "Reports on Mathematical Physics

Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems

We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction (performed with respect to a special type action of a Lie group) of micro-Jordan bihamiltonian structures whose Nijenhuis tensor has constant eigenvalues. The method is applied to the diagonal action of a Lie group $G$ on a direct product of $N$ coadjoint orbits \O=O_1\times...\times O_N endowed with a bihamiltonian structure whose first generator is the standard symplectic form on \O. As a result we get the so called classical Gaudin system on \O. The method works for a wide class of Lie algebras including the semisimple ones and for a large class of orbits including the generic ones and the semisimple ones.Comment: 24