A refinement of Ado’s Theorem

In this paper we study the minimal dimension µ(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 µ(g) , especially for nilpotent Lie algebras. Such a refinement of Ado’s theorem is required for solving a question of J. Milnor in the theory of affine manifolds. We will determine here µ(g) for certain Lie algebras and prove upper bounds in general. For nilpotent Lie algebras of dimension n, the bound nn + 1 is known. We now obtain µ(g) < α√ n 2n with some constant α ∼ 2.76287. 1