Irreducible L-modules of Modular Lie algebras

TL;DR

The paper addresses the problem of classifying irreducible modules of modular (positive characteristic) Lie algebras by leveraging the $p$-mapping and the framework of restricted and universal $p$-envelopes. It develops an induced-module approach via restricted enveloping algebras and universal $p$-envelopes, establishing correspondences between irreducible $L$-modules and suitably restricted subquotients, and introduces $S$-representations and reduced enveloping algebras to organize irreducibles by characters. The approach is illustrated through structural analysis and explicit examples (including $ ext{sl}_2$-related constructions and solvable subalgebras), culminating in a bijective understanding between irreducible modules and induced modules from subalgebras $L^{oldsymbol{ullet}}$ endowed with eigenvalue data. The work provides a practical blueprint for decomposing a modular Lie algebra into irreducibles using $p$-envelopes and induction, while highlighting open questions about extending the notion of character to broader modular settings.

Abstract

The main objective of this project is to determine all irreducible modules of a given modular Lie algebra. In contrast to ordinary Lie algebras, modular Lie algebras require an additional structure known as the p-mapping. The minimal p-envelope of a modular Lie algebra is restrictable, and there exists a one-to-one correspondence between the induced modules and certain original modules. By exploiting the properties of induced modules, this project aims to decompose a modular Lie algebra L into irreducible L- modules. Several examples will be presented to demonstrate how such decompositions can be achieved for specific modular Lie algebras.

Theorems & Definitions (67)

• Proposition 2.0.1
• proof
• Proposition 2.0.2
• proof
• Lemma 2.0.3
• proof
• Lemma 2.0.4
• proof
• Proposition 2.0.5
• proof