Locally semisimple and maximal subalgebras of the finitary Lie algebras $gl(\infty)$, $sl(\infty)$, $so(\infty)$, and $sp(\infty)$

TL;DR

This paper classifies all locally semisimple and maximal subalgebras within the infinite-dimensional finitary Lie algebras, extending classical finite-dimensional Lie algebra results to the infinite case.

Contribution

It provides a comprehensive description of subalgebras in infinite-dimensional finitary Lie algebras, a significant extension of classical finite-dimensional classifications.

Findings

All locally semisimple subalgebras characterized

All maximal subalgebras classified

Extension of finite-dimensional results to infinite case

Abstract

We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary Lie algebras $\gl(\infty), \sl(\infty), \so(\infty)$, and $\sp(\infty)$. For simple finite--dimensional Lie algebras these classes of subalgebras have been described in the classical works of A. Malcev and E. Dynkin. Key words (2000 MSC): 17B05 and 17B65.