Classification of Solvable Lie Algebras

Abstract

In this paper we illustrate some simple ideas that can be used for obtaining a classification of small-dimensional solvable Lie algebras. Using these we obtain the classification of 3 and 4 dimensional solvable Lie algebras (over fields of any characteristic). Precise conditions for isomorphism are given.Solvable Lie algebras have been classified by G. M. Mubarakzjanov (upto dimension 6 over a real field, [8], [7], see also [9]), and by J. Patera and H. Zassenhaus (upto dimension 4 over any perfect field, [10]). 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 invloved, the more systematic the methods have to be. However, the drawback is that the computer can only handle finite data. For example, below 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 I 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 1 (and for which we do not claim any originality). Then in Section 2 we describe the use of Gröbner bases. In Section 3 solvable Lie algebras of dimension 3 are classified. In Section 4 the same is done for dimension 4. We show that our classification in dimension 4 differs slightly from the one found in [10] (i.e., we find a few more Lie algebras).For doing the explicit calculations reported here we have used the computer algebra system Magma ([2]).

Classification of 4-dimensional homogeneous weakly einstein manifolds

“…Besides, it is easy to check that the curvature tensor (7.5) takes on the form Then, the space of the curvature operators is obviously spanned by the six operators A 12 , A 13 , A 14 , A 23 , A 24 , A 34 . Hence the Lie algebra generated by these operators is so (4). We see that the action of the holonomy algebra on the tangent space T e G is irreducible and hence the corresponding Riemannian manifolds are irreducible.…”

Calabi-Yau cones from contact reduction

“…This is not the only way one can obtain the Heisenberg group by means of a reduction; for example, the above construction applies equally well to the examples of Section 3 relative to the Lie group H with structure constants (12), setting X e = e 2 . On the other hand, quotients of semidirect products H ⋉ V do not generally satisfy Theorem 12 when H is solvable.…”

“…Let H be a solvable 3-dimensional Lie group. Then there exists a semidirect product H ⋉ R 4 admitting a left-invariant contact SU(3)-structure whose associated spinor is generalized Killing if and only if H has structure equations (11)(12)(13)(14).…”

“…Next we prove the "if" part of Theorem 5, by giving explicit examples for each solvable Lie group (11)(12)(13)(14).…”

Directed pseudo-graphs and lie algebras over finite fields

“…Remember that de Graff, for instance, classified solvable Lie algebras over all fields, only in dimensions 4 (see [6]), and Patera and Zassenhaus [12] classified solvable Lie algebras of dimensions 4 over perfect fields.…”