U -projection for the adjoint representation of the groupGL( n, K)

In the paper we study rings and fields of invariants for the adjoint represen- tation of the group GL(n, K) over the field of zero characteristic. The aim of this paper is to construct the special linear operator, we call it U -projector, that maps any polynomial on the matrix algebra to an U -invariant rational function. In this paper we present two different constructions of U -projector. Using the U -projector we obtain the system of generators of the field of U -invariants of the adjoint representation of the group GL(n, K). We obtain the system of genera- tors of the field of U -invariants for the restriction of the adjoint representation to the subgroup of block-diagonal matrices.

field of invariants, adjoint representation, unitriangular group, solvable group, algebra of invariants, representations of groups, locally nilpotent derivation, system of generators.