On Lie Algebras Generated by Few Extremal Elements

We give an overview of some properties of Lie algebras generated by at most 5 extremal elements. In particular, for any finite graph {\Gamma} and any field K of characteristic not 2, we consider an algebraic variety X over K whose K-points parametrize Lie algebras generated by extremal elements. Here the generators correspond to the vertices of the graph, and we prescribe commutation relations corresponding to the nonedges of {\Gamma}. We show that, for all connected undirected finite graphs on at most 5 vertices, X is a finite-dimensional affine space. Furthermore, we show that for maximal-dimensional Lie algebras generated by 5 extremal elements, X is a point. The latter result implies that the bilinear map describing extremality must be identically zero, so that all extremal elements are sandwich elements and the only Lie algebra of this dimension that occurs is nilpotent. These results were obtained by extensive computations with the Magma computational algebra system. The algorithms developed can be applied to arbitrary {\Gamma} (i.e., without restriction on the number of vertices), and may be of independent interest.Comment: 19 page

Shunting passenger trains: getting ready for departure

In this paper we consider the problem of shunting train units on a railway station. Train units arrive at and depart from the station according to a given train schedule and in between the units may have to be stored at the station. The assignment of arriving to departing train units (called matching) and the scheduling of the movements to realize this matching is called shunting. The goal is to realize the shunting using a minimal number of shunt movements.\ud For a restricted version of this problem an ILP approach has been presented in the literature. In this paper, we consider the general shunting problem and derive a greedy heuristic approach and an exact solution method based on dynamic programming. Both methods are flexible in the sense that they allow the incorporation of practical planning rules and may be extended to cover additional requirements from practice

Extremal Presentations for Classical Lie Algebras

The long-root elements in Lie algebras of Chevalley type have been well studied and can be characterized as extremal elements, that is, elements $x$ such that the image of (\ad x)^2 lies in the subspace spanned by $x$. In this paper, assuming an algebraically closed base field of characteristic not 2, we find presentations of the Lie algebras of classical Chevalley type by means of minimal sets of extremal generators. The relations are described by simple graphs on the sets. For example, for $C_n$ the graph is a path of length $2n$, and for $A_n$ the graph is the triangle connected to a path of length $n-3$.Comment: 26 pages, 6 figure

Computing Chevalley bases in small characteristics

Let L be the Lie algebra of a simple algebraic group defined over a field F and let H be a split Cartan subalgebra of L. Then L has a Chevalley basis with respect to H. If the characteristic of F is not 2 or 3, it is known how to find it. In this paper, we treat the remaining two characteristics. To this end, we present a few new methods, implemented in Magma, which vary from the computation of centralisers of one root space in another to the computation of a specific part of the Lie algebra of derivations of $L$.Comment: 22 page

Proving Statements in Planar Geometry and Cinderella

This report describes the things done in a three month internship at the Technische Universität Berlin, performed in the summer of 2003. The main goal of this internship was to make it possible to put forward a geometrical theorem (i.e. a theorem involving points, lines, conics, incidences, etc) on a computer by pointing and clicking, and then obtain a computer-generated proof for this theorem. This goal was achieved, adding various options to the computer program Cinderella[4]. With Cinderella one can create geometrical configurations, and the internal ‘Randomized prover ’ is able to discover theorems. In this internship we added the functionality to find proofs for these theorems with the aid of the computer algebra package GAP[20]. Communication between these two programs and the various steps in generating the proof is done by means of OpenMath [12, 15]. Cinderella is now able to generate a mathematically correct proof of certain theorems created by the user. Moreover, one can verify this proof without extensive knowledge of the way the proof was obtained. As this internship was performe

Automated Proofs using Bracket Algebra with Cinderella and OpenMath

This paper describes the results of a project intended to make it possible to put forward geometrical theorems by pointing and clicking, and then obtain a proof for that theorem automatically. This goal was achieved by adding various options to Cinderella [1], a computer program with which one can create geometrical configurations. Its internal ‘Randomized prover ’ is able to discover theorems automatically. In the project the functionality was added to find proofs for these theorems with the aid of the computer algebra package GAP [9]. Communication between these two programs and the various steps in generating the proof is done by means of OpenMath [5, 7]. The proofs are represented by bracket calculations as proposed in [8]

Book review: Discovering Mathematics with Magma

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