On some classes of Z-graded Lie algebras

TL;DR

This paper investigates finite-dimensional Z-graded Lie algebras, focusing on their prolongations with reductive subalgebras, providing structural decompositions and examples leading to simple Lie algebras of classical types.

Contribution

It generalizes effectiveness and algebraicity assumptions to analyze prolongations, enabling explicit decompositions and constructing examples of simple Lie algebras from nilpotent graded structures.

Findings

Established Levi-Malcev and Levi-Chevalley decompositions for these prolongations.

Provided systematic examples of simple Lie algebras as prolongations of nilpotent Z-graded Lie algebras.

Analyzed properties such as height and structural features of the prolongations.

Abstract

We study finite dimensional almost and quasi-effective prolongations of nilpotent Z-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi-Mal\v{c}ev and Levi-Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we systematically present examples in which simple Lie algebras are obtained as prolongations, for reductive structural algebras of type A, B, C and D, of nilpotent Z-graded Lie algebras arising as their linear representations.