Infinite dimensional analogues of nilpotent and solvable Lie algebras
F. H. Haydarov, B. A. Omirov, G. O. Solijanova
TL;DR
This work extends the finite-dimensional theory of nilpotent and solvable Lie algebras to infinite dimensions by developing pro-nilpotent and pro-solvable analogues, along with residually nilpotent/resolvable variants. It establishes pro-version analogues of Engel's and Lie's theorems via strict and standard triangularizability of adjoint representations, and introduces the pro-nilpotent radical and maximal residually solvable extensions, including rank and inner-derivation results in maximal cases. The authors show that maximal tori can be defined as limits of tori on finite quotients, and they develop constructions (tensor, direct sum, central extensions) that preserve pro-nilpotency, enabling systematic generation of new pro-algebras. These results connect to applications in the study of characteristic Lie algebras and PDEs by providing a robust framework for infinite-dimensional nilpotent/solvable analogues and their extensions. The paper thus broadens the toolkit for analyzing infinite-dimensional Lie algebras with controlled growth and filtration properties.
Abstract
We study infinite-dimensional analogues of nilpotent and solvable Lie algebras, focusing on the classes of pro-nilpotent, residually nilpotent, pro-solvable and residually solvable Lie algebras. We extend classical triangularization results (Engel's and Lie's theorems) to the pro-setting and establish existence results for the pro-nilpotent radical in pro-solvable algebras and in certain residually solvable algebras. We adapt finite-dimensional construction methods to produce residually solvable extensions with a given pro-nilpotent radical under natural finiteness conditions. By analyzing derivations and maximal tori of pro-nilpotent algebras, we extend the notion of rank and show that, for pro-nilpotent algebras of maximal rank, every derivation of a maximal residually solvable extension is inner. Finally, we describe standard constructions (tensor and direct sum products, central extensions) that preserve pro-nilpotency.