Nilpotent 6-dimensional Lie algebras over any field of characteristic not 2 are classified. The proof of this classification is essentially constructive: for a given 6-dimensional nilpotent Lie algebra L, following the steps of the proof, it is possible to find a Lie algebra M that occurs in the classification, and an isomorphism L → M. In the proof a method due to Skjelbred and Sund is used, along with a method based on Gröbner bases to find isomorphisms.

Let G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a representative of a nilpotent orbit. We use this to compute lists of representatives of these orbits for the Lie algebras of exceptional type. Then we give two applications. The first one concerns settling a conjecture by Elashvili on the index of centralizers of nilpotent orbits, for the case where the Lie algebra is of exceptional type. The second deals with minimal dimensions of centralizers in centralizers.

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]).

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic 2. To achieve the classification we use the action of the automorphism group on the second cohomology space, as isomorphism types of nilpotent Lie algebras correspond to orbits of subspaces under this action. In some cases, these orbits are determined using geometric invariants, such as the Gram determinant or the Arf invariant. As a byproduct, we completely determine, for a 4-dimensional vector space V , the orbits of GL(V ) on the set of 2-dimensional subspaces of V ∧ V .

Abstract. We describe a method for classifying the Novikov algebras with a given associated Lie algebra. Subsequently we give the classification of the Novikov algebras of dimension 3 over R and C, as well as the classification of the 4-dimensional Novikov algebras over C whose associated Lie algebra is nilpotent. In particular this includes a list of all 4-dimensional commutative associative algebras over C.

Abstract. We classify nilpotent associative algebras of dimensions up to 4 over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras.

Algorithms are described that help with obtaining a classification of the semisimple subalgebras of a given semisimple Lie algebra, up to linear equivalence. The algorithms have been used to obtain classifications of the semisimple subalgebras of the simple Lie algebras of ranks 8. These have been made available as a database inside the SLA package of GAP4. The subalgebras in this database are explicitly given, as well as the inclusion relations among them.

It is well known that a Severi-Brauer surface has a rational point if and only if it is isomorphic to the projective plane. Given a Severi-Brauer surface, we study the problem to decide whether such an isomorphism to the projective plane, or such a rational point, does exist; and to construct such an isomorphism or such a point in the affirmative case. We give an algorithm using Lie algebra techniques. The algorithm has been implemented in Magma.