Bull. Kyushu Inst. Tech.
(Math. Natur. Sci,) No. 34, 1987, pp. 5-8
NOTES ON INVARIANT RADICALS OF LOCALLY
FINITE DERIVATIONS
By Fujio KuBo
(Received November 29, 1986)
Introduction
For a Lie algebra L over an algebraically closed field f of characteristic zero, Krempa shows in [5] that every subspace invariant under all automorphisms of L is also invariant under all locally finite derivations of L. From this result we can easily derive the propo- sition that for a Lie algeie.ra L given above, A which is one of the relations si, asc, desc, lsi, ser and for any class X, every subalgebra generated by all zdsc-subalgebras of L is invariant under all locally finite derivations of L. In this paper we present some classes se for which the above proposition holds for a locally fini'te Lie algebra L without the assumption that f is algebraically closed.
1. Classes and radicals
A class Xf is a collection of Lie algebras over a field f together with their isomorphic copies and the O-dimensional Lie algebra. A class sc is said to be linear if LE eet implies L(g)f S2Ealg for any extension field S;2 of f, and if L(g)f S2Eaig for some extension field 9 offimplies LE Xf (cf. Jacobson [3; p. 146]).
Let f be a field of characteristic zero, 2 be an extqnsion field of { and G(91b be the group of all f-automorphisms of 9. We call ec G(91b-invariant if for a Lie algebra L over f and a subalgebra H of L(g)f 9 with He ecQ, (1(g)g)(H) belongs to ceg for any gE G(S2/b.
We shall give some classes which are linear and G(9/b-invariant, so called the varie- ties: Let F. be the free Lie algebra over a field e of rational numbers on an infinite set {xi, x2,...}, Fi be the free Lie algebra over e on {xi,•••,xi}• Let s=Z.,.a.,.[x.(i),•••, x.(.)] be an element of F., where o(i) is a natural number Iarger than or equal to 1 and a.,.EC. An element s is called multilinear if the numbers of times that xi occurs in [x.(i),•••, x.(.)] are same and they are O or 1. Let M be the set of all multilinear elements of F.. Let S be a subset of M, The variety 8s corresponding to S is a class of all Lie algebras L such that s(ei,.,., e.)=O for any sES with sEF. and for any ei,..., e.GL. For instance, if S={[xi,..., x,+i]}, then 8s is the class of all nilpotent Lie algebras of class
5c (Amayo and Stewart [1; Chap, 14]).
Let se=8s be a variety. Since every element of S is multilinear, X is obviously linear.
Let L be a Lie algebra over f and H be a subalgebra of L(g)f S2 with HE2Eg. Take sES with s= s(xi,..., x.) E F.. Since (1 (g) g)( [u, v]) = [(1 (g) g)(u), (1 (g) g)(v)] for any u, ve
L (g) , S2 and g E G( S2/b, s((1 (8) g)(h,),.,., (1 (g) g)(h.)) == O for any h,,..., h. E H. Therefo re we have (1(g)g)(H)ece., which shows that sc is G(9!b-invariant. It is clear that the union of ljnear classes invariant of G(9/{) is also linear and G(9!b-invariant. Hence the class ER of all nilpotent Lie algebras holds these two conditions, similarly so does the class ES2I of all solvable Lie algebras. We notice that the class 89 of all finite-dimensional Lie algebras satisfies these two conditions.
Every Lie algebra considered in this paper will be over a field of characteristic zero, and we shall use the notations and terminology of [1]. In particular `Åq ', `si', `asc',
`desc', `lsi', and `ser' denote the relations `ideal', `subideal', `ascendant subalgebra',
`descendant subalgebra', `local subideal', and `serial subalgebra' respectively.
Let A be any relation between a Lie algebra and its subalgebra, for example, si, asc, lsi, ser. A class 2E is A-coalescent if A, BE sc, AAL and BAL then ÅqA, BÅr E se and ÅqA, BÅrA L. For instance, itg, IYnEn and IYnEQ( are A-coalescent for A==si, asc, lsi (Hartley [2], Amayo & Stewart [1; pp. 257, 263]).
For zl given above, a class a; and a Lie algebra L, .IAor(L) is defined as J, re(L) = ÅqHAL: H E XÅr ,
We notice that J,iee(L)gJ.,.ee(L)gJ,.,es(L) and J,ies(L)gJi,ire(L)•
PRoposmoN 1. Let XgEg and L be a locally finite Lie algebra. Then Ji,ift-(L)=
Jserce(L)•
PRooF. It is enough to show that for a finite-dimensional subalgebra H of L, H lsi L ifand only ifHserL. Kashiwagi shows in [4] that HlsiLimplies HserL. Conversely assume that HserL. Then by [1;Proposition13.2.4], HnKsiK for any finite- dimensional subalgebra K of L. Now let X be any finite subset of L. Since ÅqH, XÅr is finite-dimensional,H =Hn ÅqH, XÅr sj ÅqH, XÅr. This shows thatH lsi L. Q. E. D.
ExAMpLE 1 : Let f be a field ofcharacteristic zero and P=f[t] be a polynomial algebra.
Considered as an abelian Lie algebra P has derivations x: pHtp and y: pHdpldt. Then [x, y] ==z is the identity on P. Let L be the split extension P+Åqx, y, zÅr (the Hartley algebra [1; Example 6.3.6]). For this Lie algebra L, we have the following relations of inclusions :
Jsig(L)
ll E Jascg(L) EIii Jserrv(L){
Jisig(L)
'
