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

Classification of filiform Lie algebras up to dimension 7 over finite fields

This paper tries to develop a recent research which consists in using Discrete Mathematics as a tool in the study of the problem of the classification of Lie algebras in general, dealing in this case with filiform Lie algebras up to dimension 7 over finite fields. The idea lies in the representation of each Lie algebra by a certain type of graphs. Then, some properties on Graph Theory make easier to classify the algebras. As main results, we find out that there exist, up to isomorphism, six, five and five 6-dimensional filiform Lie algebras and fifteen, eleven and fifteen 7-dimensional ones, respectively, over Z/pZ, for p = 2, 3, 5. In any case, the main interest of the paper is not the computations itself but both to provide new strategies to find out properties of Lie algebras and to exemplify a suitable technique to be used in classifications for larger dimensions

Cohomology of abelian matched pairs and the Kac sequence

AbstractThe purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite groups and finite-dimensional Lie algebras. The morphisms in the low degree part of this sequence are given explicitly, enabling concrete computations

Affine.m - Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

In this paper we present Affine.m - program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. Algorithms are based upon the properties of weights and Weyl symmetry. The most important problems for us are the ones, concerning computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras.Comment: 29 pages, 7 figures, updated to match published versio

Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space

This work is essentially a detailed version of notes published by the author (some in collaboration) in the “Comptes Rendus Hebdomadaires des Séances de l’Académie de Sciences, Série A (Paris)” in 1971 and early 1972 (numbers 272 and 274). The first chapter studies Lie algebras constituted by finite rank operators in a complex Hilbert space, and which correspond in infinite dimensions to the classical simple Lie algebras in finite dimensions. Completions of these with respect to the Schatten uniform crossnorms and to the weak topology provide the Banach-Lie algebras investigated in chapter II. As applications : homogeneous spaces of the corresponding Banach-Lie groups are studied in chapter III; the relationship between cohomologies of the groups, of their Lie algebras and of their classifying spaces is the subject of chapter IV. The main results are: Algebras of derivations, groups of automorphisms, classification of the real forms and cohomological computations for the Lie algebras introduced in chapters I and II. The methods used have been suggested by the theories of finite dimensional semi-simple real and complex Lie algebras, and of associative Banach algebras of linear operators. The results are related to the theory of L*-algebras (Schue, Balachandran) and to the investigations about the Lie structure of simple associative rings (Herstein, Martindale)

Commutative 2-cocycles on Lie algebras

On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including finite-dimensional semisimple, current and Kac-Moody algebras.Comment: v7: minor changes; added ancillary file with GAP cod

Dimensions of Imaginary Root Spaces of Hyperbolic Kac--Moody Algebras

We discuss the known results and methods for determining root multiplicities for hyperbolic Kac--Moody algebras