A note on the Lie algebras of algebraic groups

Abstract

0. In his book [1] C. Chevalley defined the replicas for any elements of Lie algebras of algebraic groups of matrices which are defined over fields of characteristic 0, and he characterized algebraic subalgebras as those subalgebras of the general linear algebras which are closed with respect to "replica operation" i. e. those which contain all replicas of any elements of themselves. In this paper we shall define the replica in the case of any algebraic groups defined over fields of characterestic 0 and show that the same characterization of algebraic subalgebras is true in this case too.1. Let G be a connected algebraic group 0

A certain class of commutative algebraic groups

“…Since H normalizes M, for any x of H, Ad(x) maps m into itself, where Ad means the adjoint representation of G; and therefore the corollary of the proposition 4 of [3] shows that ΐ) normalized m; thus the subalgebra of g generated by ί) and m is ί) + m. By the main theorem of [3] we have that ΐ) + m is the Lie algebra of the connected algebraic group HM. Thus the proof of Lemma 1 is complete.…”

“…Recently the author constructed the Lie theory of algebraic groups in his papers [3] and [4] where a certain class of algebraic groups played an important part let G be a connected algebraic group defined over a field of characteristic 0; let g be the Lie algebra of G then it is shown that for any D of g there exists the minimal algebraic subgroup G(D) of G whose Lie algebra contains Zλ This algebraic subgroup G(D) corresponds to the closed subgroup generated by the one-parameter subgroup determined by the element D of Lie algebra in the classical theory of Lie groups. In the linear case C. Chevalley has obtained the structural properties of this class of algebraic groups G(D): for any X of $t(n, k), the subgroup G(X) of GL(n, k) is commutative; if t is an indeterminate, the point exp tX is a generic point over k on G(X) where k is a field of characteristic 0.…”

“…The converse is true without the condition of the characteristic: in fact, if G is commutative, the adjoint representation Ad(G) of G reduces to the unit element and the Lie algebra of Ad(G) does to the zero matrix; then the propoŝ ition 1 of [3] shows that g is commutative. This method is orthodoxy Rosenlicht [5] gave an interesting proof, using the invariant differential forms on G. This theorem is the generalization of the result which is stated in the theorem 10 of the chapter Π of [1].…”