Faithful completely reducible representations of modular Lie algebras

TL;DR

This paper strengthens Jacobson's theorem by proving that any finite-dimensional modular Lie algebra has a faithful completely reducible module with dimension bounded by a function of its dimension and the characteristic.

Contribution

It provides an explicit upper bound on the dimension of faithful completely reducible modules for modular Lie algebras, improving previous existence results.

Findings

Established a bound of p^{n^2-1} on the dimension of faithful modules

Extended Jacobson's theorem to include explicit dimension bounds

Confirmed the existence of such modules with the given size constraint

Abstract

The Ado-Iwasawa Theorem asserts that a finite-dimensional Lie algebra $L$ over a field $F$ has a finite-dimensional faithful module $V$. There are several extensions asserting the existence of such a module with various additional properties. In particular, Jacobson has proved that if the field has characteristic $p>0$, then there exists a completely reducible such module $V$. I strengthen Jacobson's Theorem, proving that if $L$ has dimension $n$ over the field $F$ of characteristic $p>0$, then $L$ has a faithful completely reducible module $V$ with $\dim(V) \le p^{n^2-1}$.