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 ${\alpha \sim 2.76287}$.