# Donald Barnes (University of Sydney)

## Faithful completely reducible representation of modular Lie algebras

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 $$V$$ with various additional properties. In particular, Jacobson has proved that if the field $$F$$ has characteristic $$p > 0$$, then there exists a completely reducible such module $$V$$ . I prove that if $$L$$ is of dimension $$n$$ over $$F$$ of characteristic $$p$$, then $$L$$ has a faithful completely reducible module $$V$$ with $$dim(V ) \le p^{n^2-1}$$.