ИСТИНА

Let us consider a Lie algebra g. On its dual space g* the Lie-Poisson bracket Ax and the Poisson bracket with constant coefficients Aa are naturally defined. The Jordan-Kronecker decomposition theorem gives the classification of pairs of skewsymmetric bilinear forms by reducing them simultaneously to a canonical blockdiagonal form. For any two points x, a ∈ g* there exists a pencil of bilinear forms Ax − λAa at the point x. The sizes of blocks in the Jordan-Kronecker decomposition of Ax and Aa are called the algebraic type of a pencil. For almost all pencils their algebraic types are the same. So we can call the algebraic type of a common pencil the Jordan-Kronecker invariants of a Lie algebra. A.Yu. Groznova calculated the Jordan-Kronecker invariants for nilpotent Lie algebras with dimension seven in her diploma work. On the other hand, the article written by Alfons I. Ooms is concerned with studying the properties of the rings of coajoint invariants for the same Lie algebras. Groznova’s calculations led to the conclusion that the existence of Kronecker pencils of different algebraic types but of the same rank can be a characteristic property for the non-free generatedness of the ring of coajoint invariants for Kronecker nilpotent Lie algebras. So the problem was to find such pencils for every 7-dimensional algebra with Kronecker type and non-free generated ring of coajoint invariants. Thus the algorithm for finding these pencils or proving that they cannot exist will be presented. Also we will discuss the full list of the results of this algorithm work.