# Polynomially and Infinitesimally Injective Modules

## TL;DR

This paper investigates the conditions under which polynomial injective modules for a general linear group are also injective over the restricted enveloping algebra, providing explicit results for the case when n=2.

## Contribution

It characterizes polynomially injective modules that are also injective over the restricted enveloping algebra, with explicit results for the case n=2.

## Key findings

- Explicit characterization for n=2
- Connection between polynomial injectivity and restricted enveloping algebra injectivity
- Analysis of divisibility index of polynomial modules

## Abstract

The injective polynomial modules for a general linear group $G$ of degree $n$ are labelled by the partitions with at most $n$ parts. Working over an algebraically closed field of characteristic $p$, we consider the question of which partitions correspond to polynomially injective modules that are also injective as modules for the restricted enveloping algebra of the Lie algebra of $G$. The question is related to the "index of divisibility" of a polynomial module in general, and an explicit answer is given for $n=2$.