Dan Roozemond

Simple Lie algebras having extremal elements

Let L be a simple finite-dimensional Lie algebra of characteristic distinct from 2 and from 3. Suppose that L contains an extremal element that is not a sandwich, that is, an element x such that [x, [x, L]] is equal to the linear span of x in L. In this paper we prove that, with a single exception, L is generated by extremal elements. The result is known, at least for most characteristics, but the proofs in the literature are involved. The current proof closes a gap in a geometric proof that every simple Lie algebra containing no sandwiches (that is, ad-nilpotent elements of order 2) is in fact of classical type.

Computing Chevalley bases in small characteristics

Abstract Let L be the Lie algebra of a simple algebraic group defined over a field F and let H be a split maximal toral subalgebra of L. Then L has a Chevalley basis with respect to H. If char ( F ) ≠ 2 , 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 centralizers of one root space in another to the computation of a specific part of the Lie algebra of derivations of L.

Symbolic Computation Software Composability

We present three examples of the composition of Computer Algebra Systems to illustrate the progress on a composability infrastructure as part of the SCIEnce (Symbolic Computation Infrastructure for Europe) project. One of the major results of the project so far is an OpenMath based protocol called SCSCP (Symbolic Computation Software Composability Protocol). SCSCP enables the various software packages for example to exchange mathematical objects, request calculations, and store and retrieve remote objects, either locally or accross the internet. The three examples show the current state of the GAP, KANT, and MuPAD software packages, and give a demonstration of exposing Macaulay using a newly developed framework.

Easy composition of symbolic computation software using SCSCP: A new Lingua Franca for symbolic computation

We present the results of the first four years of the European research project SCIEnce-Symbolic Computation Infrastructure in Europe (http://www.symbolic-computing.org), which aims to provide key infrastructure for symbolic computation research. A primary outcome of the project is that we have developed a new way of combining computer algebra systems using the Symbolic Computation Software Composability Protocol (SCSCP), in which both protocol messages and data are encoded in the OpenMath format. We describe the SCSCP middleware and APIs, outline implementations for various Computer Algebra Systems (CAS), and show how SCSCP-compliant components may be combined to solve scientific problems that cannot be solved within a single CAS, or may be organised into a system for distributed parallel computations. Additionally, we present several domain-specific parallel skeletons that capture commonly used symbolic computations. To ease use and to maximise inter-operability, these skeletons themselves are provided as SCSCP services and take SCSCP services as arguments.

Easy composition of symbolic computation software: a new lingua franca for symbolic computation

We present the results of the first four years of the European research project SCIEnce (www.symbolic-computation.org), which aims to provide key infrastructure for symbolic computation research. A primary outcome of the project is that we have developed a new way of combining computer algebra systems using the Symbolic Computation Software Composability Protocol (SCSCP), in which both protocol messages and data are encoded in the OpenMath format. We describe SCSCP middleware and APIs, outline some implementations for various Computer Algebra Systems (CAS), and show how SCSCP-compliant components may be combined to solve scientific problems that can not be solved within a single CAS, or may be organised into a system for distributed parallel computations.

Extremal presentations for classical Lie algebras

Abstract 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 .

Computing split maximal toral subalgebras of Lie algebras over fields of small characteristic

Important subalgebras of a Lie algebra of an algebraic group are its toral subalgebras, or equivalently (over fields of characteristic 0) its Cartan subalgebras. Of great importance among these are ones that are split: their action on the Lie algebra splits completely over the field of definition. While algorithms to compute split maximal toral subalgebras exist and have been implemented (Ryba, 2007; Cohen and Murray, 2009), these algorithms fail when the Lie algebra is defined over a field of characteristic 2 or 3. We present heuristic algorithms that, given a reductive Lie algebra L over a finite field of characteristic 2 or 3, find a split maximal toral subalgebra of L. Together with earlier work (Cohen and Roozemond, 2009) these algorithms are very useful for the recognition of reductive Lie algebras over such fields.

On Lie algebras generated by few extremal elements

Abstract We give an overview of some properties of Lie algebras generated by at most 5 extremal elements. In particular, for any finite graph Γ 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 Γ . 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 single 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 Γ (i.e., without restriction on the number of vertices), and may be of independent interest.