SYMMETRY ANALYSIS OF THE CANONICAL CONNECTION ON LIE GROUPS:CO-DIMENSION TWO ABELIAN NILRADICAL WITH ABELIAN AND NON ABELIAN COMPLEMENT

Abstract

We consider the symmetry algebra of the geodesic equations of the canonical connection on a Lie groups. We mainly consider the solvable indecomposable four, five and six-dimensional Lie algebras with co-dimension two abelian nilradical, that have an abelian and not abelian complement. In this particular case, we have only one algebra in dimension four namely; A4,12 , and three algebras in dimension five namely; A5,33, A5,34, and A5,35 In dimension six, based on the list of Lie algebras in Turkowski’s list, there are nineteen such algebras namely; A6,1- A6,19 that have an abelian complement, and there are eight algebras that have a non-abelian complement namely; A6,20- A6,27. For each algebra, we give the geodesic equations, a basis for the symmetry Lie algebra in terms of vector fields. Finally we examine each case and identify the symmetry Lie algebra.