Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal and Geometry of Coadjoint Orbits of Corresponding Lie Groups

Abstract

Given a Lie algebra G , let µ(G) be the minimal degree of a faithful representation of G . This is an integer valued invariant of G , which has been introduced by D. Burde in 1998. It is not known, in general, how to determine this invariant for a given solvable Lie algebra. Lie (n, k) denotes the class of all n-dimensional real solvable Lie algebras having k -dimensional derived ideals. In 2020 we gave a classification of all non 2-step nilpotent Lie algebras of Lie (n, 2). We propose in this paper to study representations of these Lie algebras. First, we give an upper bound of µ(G) for each G classified. For indecomposable case, we further compute the character afforded by the adjoint representation and describe the picture of coadjoint orbits of Lie groups corresponding to them.