On the Algebraic Hull of a Lie Algebra

Let F be a field of characteristic 0, and let F be a finite dimensional vector space over F. Let E denote the algebra of all endomorphisms of V, and let L be any Lie subalgebra of E. Among the algebraic Lie algebras contained in E and containing L, there is one that is contained in all of them, and this is called the algebraic hull of L in E. Here, an algebraic Lie algebra is defined as the Lie algebra of an algebraic group. It is an easy consequence of the definitions that if A and B are algebraic groups of automorphisms of V such that AÇ_B then the Lie algebra of A is contained in the Lie algebra of B. Hence the existence of the algebraic hull of L is an immediate consequence of the following basic result: let G be the intersection of all algebraic groups of automorphisms of V whose Lie algebras contain L. Then the Lie algebra of G contains L. This theorem reduces at once to the case where L is one dimensional. For any xGE, let Gx be the intersection of all algebraic groups of automorphisms of V whose Lie algebras contain x. Then we have GxdG, whenever xG-G and it suffices to show that the Lie algebra of Gx contains x. This is part of [l, Theorem 10, p. 165], but it is not clear from the proof given in [l] that, although this result is not as obvious as it might seem at first sight, it can be proved quite directly without invoking any special knowledge of algebraic groups. The proof we give here is based on the simple idea of recovering Gx from its generic point exp(/x). We use an auxiliary variable / over F and introduce the ring F\t\ of the integral power series in t with coefficients in F. Let E* denote the dual space of E, and let P be the algebra of all polynomial functions on E. The elements of E* are canonically extended to become F {t} -linear maps of E ® FF [ t} into F {t}, and the elements of P are extended accordingly to become F {t} -valued functions on E ® pF {t}. With x(E.E, we interpret the formal power series exp(/x) as the element of E®pF{t\ that is determined by the conditions