Engel's Theorem in generalized lie algebras

Abstract

In this thesis we deal with Engel's Theorem about simultaneous triangulability of the space of nilpotent operators closed under Lie bracket, one of the corner stones of Lie Theory. This theorem was first proven in 1892 by F. Engel in his paper [4]. Since then several various versions of this theorem and its proofs have been suggested ([3], [8], [12]). In some versions the authors deal with weakly closed sets of elements in associative algebras [6]. In the others they look at representations of Lie algebras by nilpotent transformations of vector spaces [3]. -- Recently people began looking at the version of Engel’s Theorem for generalized Lie algebras. Engel’s Theorem in the case of ordinary Lie superalgebras was mentioned (without proof) in the fundamental paper of V. Kac [8] devoted to the classification of simple finite-dimensional Lie superalgebras and in the monograph of M. Scheunert [12]. It was quite clear that a similar result should hold also in the case of more general color Lie superalgebras [2]. The most recent development leads to Lie algebras over Hopf algebras. A version of Engel;s Theorem for this much more general setting was suggested in the Ph. D. dissertation of V. Linchenko [9]. -- In this dissertation we choose one of the possible versions of Engel’s Theorem, in the spirit of Bourbaki [3], using the approach via representation theory. We demonstrate how this approach can work in the case of the color Lie superalgebras. We also tried the case of so called (H,β)-Lie algebras where β is a bicharacter on a cotriangular Hopf algebra H. The result we give here generalizes the case of ordinary Lie algebras but when restricted to the case of color Lie superalgebras produces a considerably weaker result. The proof of this result and several complementary lemmas was communicated to us by M. Kotchetov.