Classification of Solvable Lie Algebras

Several classifications of solvable Lie algebras of small dimension are known. Up to dimension 6 over a real field they were classified by G. M. Mubarakzjanov [Mubarakzjanov 63a, Mubarakzjanov 63b], and up to dimension 4 over any perfect field by J. Patera and H. Zassenhaus [Patera and Zassenhaus 90]. In this paper we explore the possibility of using the computer to obtain a classification of solvable Lie algebras. The possible advantages of this are clear. The problem of classifying Lie algebras needs a systematic approach, and the more the computer is involved, the more systematic the methods have to be. However, the drawback is that the computer can only handle finite data. For example, we will consider orbits of the action of the automorphism group of a Lie algebra on the algebra of its derivations. Now, if the ground field is infinite, then we know of no algorithm for obtaining these orbits. In our approach we use the computer (specifically the technique of Gröbner bases) to decide isomorphism of Lie algebras, and to obtain explicit isomorphisms if they exist. The procedure that we use to classify solvable Lie algebras is based on some simple ideas, which are described in Section 2 (and for which we do not claim any originality). Then in Section 3 we describe the use of Gröbner bases for obtaining isomorphisms. In Section 4 solvable Lie algebras of dimension 3 over any field are classified. In Section 5 the same is done for dimension 4. We show that our classification in dimension 4 differs slightly from the one found in [Patera and Zassenhaus 90] (i.e., we find a few more Lie algebras).