A historical review of the classifications of Lie algebras (Q2871818)

The purpose of this rather interesting paper is to provide a short description of the current situation in the classification problem of Lie algebras, as well as of various of the different approaches developed recently and currently in progress. Albeit for obvious reasons it is nowadays impossible to summarize all existing work in the classification problem, the authors succeed in giving a valuable overview to the beginner or the expert not directly devoted to the classification problem. It should be added that such reviews for the classification of Lie algebras are rather scarce, hence this paper constitutes a welcome addition.NEWLINENEWLINE NEWLINEIn the first section an historical introduction to the classification problem is given, with the pioneering work of Lie, Killing, Engel, Umlauf and Cartan. The case of finite dimensional Lie algebras is presented first. The various hierarchies of Lie algebras (simple, semisimple, solvable, semidirect sums of semisimple and solvable, nilpotent) are discussed, with emphasis on the differences and validity of characterizations depending on the characteristic of the base field. Special attention is devoted to the class of (real and complex) nilpotent Lie algebras, which structurally represent the main (and insurmountable) obstruction to obtain complete classifications for higher dimensions. This impossibility (at least with the currently known sets of invariants) follows from the analysis of metabelian Lie algebras developed in [\textit{L. Yu. Galitski} and \textit{D. A. Timashev}, J. Lie Theory 9, No. 1, 125--156 (1999; Zbl 0923.17015)]. This obstruction naturally leads to consider nilpotent Lie algebras subjected to some additional constraint, which can be of geometrical, algebraic or cohomological nature. Among the different types enumerated we find the naturally graded, the characteristic nilpotent or the admissible nilpotent Lie algebras. Once the main results in this direction have been reviewed, the authors devote to the problem (currently in progress) of determining the isomorphism classes of solvable Lie algebras with a nilradical of a given specific type. Most of these extensions are motivated by applications, such as the construction of integrable systems or (pseudo-Riemannian) manifolds endowed with supplementary structure.NEWLINENEWLINE A second part is devoted to review the classification of infinite dimensional Lie algebras (Kac-Moody, Cartan-type nilpotent), as well as interesting problems derived from this classification. I only miss some more detailed comments concerning the Lie algebras of vector fields, such as developed by \textit{D. B. Fuks} [Cohomology of infinite-dimensional Lie algebras. New York: Consultants Bureau (1986; Zbl 0667.17005)]. The various generalizations of Lie algebras (superalgebras, \(\mathbb{N}\)-graded Lie algebras) are also briefly reviewed.NEWLINENEWLINE The authors emphasize the importance that computer methods have gained in the classification problem, commenting on various of the techniques employed to reduce or simplify the distinction of isomorphism classes when the traditional procedures are no more applicable. It is expected that improvements of these methods will enable additional classification of special classes of Lie algebras in the coming years.NEWLINENEWLINE An extensive reference list containing 172 items provides not only the classical and canonical articles on the classification of Lie algebras, but also many recent references addressing to particular types of Lie algebras, thus providing an excellent overview on the different topics that are currently under scrutiny. The list also contains some rare but quite influential papers on the classification problem. For the beginner in the topic of classification problems, this list constitutes an extraordinary source to get familiar with the different techniques used and the problems that have still not been satisfactorily solved.