JORDAN-KRONECKER INVARIANTS FOR LIE ALGEBRAS OF SMALL DIMENSIONS

In this paper, Jordan-Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space g proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A - [lambda][BETA])([chi]), we consider Jordan-Kronecker invariants for this pencil and how they change with [chi] (the latter is done for 6-dimensional Lie algebras).
1. Basic Definitions and Theorems Definition 1. A Poisson bracket is a bilinear skew-symmetric operation over functions f,g [right arrow] {f,g} satisfying the Jacobi identity and the Leibniz rule. The [...]