The structure of almost Abelian Lie algebras

Abstract

An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media - cosmology, crystallography etc. In differential geometry and theoretical physics, almost Abelian Lie groups have given rise to some of the simplest solvmanifolds on which various geometric structures such as symplectic, K\"ahler, spin etc., are currently studied in explicit terms. However, a systematic study of almost Abelian Lie groups and algebras from mathematics perspective has not been carried out yet, and the present paper is the first step in addressing this wide and diverse class of groups and algebras. The present paper studies the structure and important algebraic properties of almost Abelian Lie algebras of arbitrary dimension over any field of scalars. A classification of almost Abelian Lie algebras is given. All Lie subalgebras and ideals, automorphisms and derivations, Lie orthogonal operators and quadratic Casimir elements are described exactly.