Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal

Tu T. C Nguyen, Vu A. Le

Published 2021-05-02Version 1

Given a Lie algebra $\mathcal{G}$, let $\mu(\mathcal{G})$ be the minimal degree of a faithful representation of $\mathcal{G}$. This is an integer valued invariant of $\mathcal{G}$, which has been introduced by D. Burde in 1998. In general it is not known how to determine this invariant. In particular, this seems to be very hard 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). In this paper, we give an upper bound of $\mu(\mathcal{G})$ for each $\mathcal{G}$ classified. For indecomposable case, we further compute the characters afforded by the adjoint, coadjoint representations and describe the picture of coadjoint orbits of Lie groups corresponding to them.