Automorphism Groups of Nilpotent Lie Algebras (original) (raw)

1987, Journal of the London Mathematical Society

In [2], the author and J. R. J. Groves showed that every linear algebraic group over a field of characteristic 0 arises from some nilpotent Lie algebra as the group of linear transformations induced on the commutator quotient by the automorphism group of the algebra. The purpose of the present paper is to extend this result to apply to arbitrary fields. Let L be a finitely generated nilpotent Lie algebra over a field k, and let m be the dimension of L. Choosing a basis of L, we may regard the automorphism group Aut (L) as a subgroup of GL (m, k). Let K be an algebraically closed extension field of k. Then K® L (tensor product taken over k) has the structure of a Lie algebra over K. Taking the basis of K® L corresponding to the chosen basis of L we may regard Aut (K ® L) as a subgroup of GL (m, K), and we have Aut (L) = Aut (K ® L) n GL (m, k). As remarked in [2], Aut(K® L) is a fc-closed subgroup of GL(m, K), that is, a subgroup closed in the Zariski fc-topology. Thus (see [4, p. 217]) Aut (K®L) is defined over some field K, where K is a finite purely inseparable extension of k and k £ K £ K. Let L denote the derived algebra of L, that is, L = [L, L], (We shall use commutator notation for Lie products.) Then Aut(L) induces on L/L' a group of linear transformations which we denote by Aut* (L). Choosing a basis of L/L, we may regard Aut*(L) as a subgroup of GL in, k), where n = dim (L/L'). Similarly, with the corresponding basis of (K® L)/(K® L)', Aut* (K (g) L) is a subgroup of The natural map from L to L/L' gives rise to a homomorphism : Aut OK L) > GL (n, K) with image Aut* (K ® L). It is easy to verify that 0 is a morphism of algebraic groups defined over K. Therefore (see [4, p. 218]) Aut* (K® L) is an algebraic subgroup of GL(n, K) defined over K. In particular, Aut* (K® L) is /c-closed. If char(/:) = 0, then K = k, and so Aut* (A^ ® L) is A:-closed; while if char (k) = p # 0, then Aut* (K ® L) is jc'-closed for every power q of p (by an easy argument), and so again Aut*(AT® L) is ^-closed. If char (k) = 0 then, as noted in [2], Aut* (L) = Aut* (K® L) D GL in, k), and consequently Aut*(L) is a Zariski-closed subgroup of Gh(n,k). The latter statement also holds if k is algebraically closed (by the discussion above with K=k)