Induced modules for modular Lie algebras

TL;DR

This paper constructs an induced module functor for finite-dimensional modules over a finite-dimensional Lie algebra in positive characteristic, establishing a categorical adjunction between modules over a subalgebra and the whole algebra.

Contribution

It introduces a new categorical framework for inducing modules from subalgebras to the entire Lie algebra in the modular setting.

Findings

Existence of a category of modules with an adjoint induction functor.

Establishment of a left adjoint to the restriction functor.

Framework applicable to finite-dimensional modules over modular Lie algebras.

Abstract

Let $L$ be a finite-dimensional Lie algebra over a field of non-zero characteristic and let $S$ be a subalgebra. Suppose that $X$ is a finite set of finite-dimensional $L$-modules. Let $D$ be the category of all finite-dimensional $S$-modules. Then there exists a category $C$ of finite-dimensional $L$-modules containing the modules in $X$ such that the restriction functor Res$:C \to D$ has a left adjoint Ind$:D \to C$.