Connection between the ring of Ad*-invariant polynomials and the Jordan–Kronecker invariants of nilpotent low-dimensional Lie

Abstract

This article is concerned with the study of connections between the Jordan–Kronecker invariants and free generatedness of the ring of Ad*-invariant polynomials of Lie algebras of dimension less than or equal to seven. At the dual space of the Lie algebra it is possible to define the Poisson bracket with the constant coefficients and the Lie-Poisson bracket. Thus, any pair of points from this dual space defines an one-parameter family of skew-symmetric bilinear forms, called a pencil. For any two bilinear forms from the pencil there exists a basis, in which
their matrices can be simultaneously reduced to the block-diagonal form with the blocks of two types. This form is called the Jordan-Kronecker decomposition. At the same time, the number and sizes of blocks will be the same for any pair of bilinear forms from the pencil. The algebraic type of a pencil is the number and sizes of blocks in the Jordan-Kronecker decomposition of any pairs of bilinear forms from the pencil. Almost all pencils of the same Lie algebra have the same algebraic type, which is the Jordan-Kronecker invariant of a given Lie algebra. There is a theorem that states that for a nilpotent Lie algebra, the existence of two Kronecker pencils of the same rank but of different algebraic types means that the ring of Ad*-invariant polynomials
must be non-freely generated. In this paper, we considered all Kronecker Lie algebras (from the certain list of 7-dimensional nilpotent Lie algebras) for which there was a possibility of the existence of a Kronecker pencils of the same rank as the rank of the algebra. As a result of the research, a negative answer was obtained to the question of whether the converse statement to the previous theorem is true.