Classification of solvable Lie algebras with a given nilradical by means of solvable extensions of its subalgebras

Abstract

We construct all solvable Lie algebras with a specific n –dimensional nilradical n n , 3 which contains the previously studied filiform ( n - 2 ) –dimensional nilpotent algebra n n - 2 , 1 as a subalgebra but not as an ideal. Rather surprisingly it turns out that the classification of such solvable algebras can be deduced from the classification of solvable algebras with the nilradical n n - 2 , 1 . Also the sets of invariants of coadjoint representation of n n , 3 and its solvable extensions are deduced from this reduction. In several cases they have polynomial bases, i.e. the invariants of the respective solvable algebra can be chosen to be Casimir invariants in its enveloping algebra.