Periodic Lie Modules

TL;DR

This paper investigates the periodicity of Lie modules in characteristic p, revealing their Heller translates and establishing their periods depending on whether p is odd or equal to 2.

Contribution

It provides a detailed description of the Heller translates of the periodic Lie module and determines its exact period in different cases.

Findings

Period of Lie modules is 2p-2 for odd p.

Period of Lie modules is 1 for p=2.

Heller translates of Lie modules are explicitly described.

Abstract

Let $p$ be a prime number and $k$ be a positive integer not divisible by $p$. We describe the Heller translates of the periodic Lie module $\mathrm{Lie}(pk)$ in characteristic $p$ and show that it has period $2p-2$ when $p$ is odd and $1$ when $p=2$.