On a refinement of Ado's theorem

Abstract

In this paper we study the minimal dimension $ \mu (g) $ of a faithful g-module for n-dimensional Lie algebras g. This is an interesting invariant of g which is difficult to compute. It is desirable to obtain good bounds for $ \mu (g) $ , especially for nilpotent Lie algebras. We will determine here $ \mu (g) $ for certain Lie algebras and prove upper bounds in general. For nilpotent Lie algebras of dimension n, the bound n n + 1 is known. We now obtain $ {\mu ({g})\le{\alpha \over \sqrt {n}}2^n} $ with some constant $ {\alpha \sim 2.76287} $ .