Bull. Kyushu Inst. Tech.
(Math. Natur. Sci.) No. 38, 1991, pp. 23-30
INFINITE-DIMENSIONAL LIE ALGEBRAS WITH NULL JACOBSON RADICAL
By Fujio KuBo
(Received January 21, 1991)
O. Introduction
For a Lie algebra Lthe Jacobson radical of Lis defined to be the intersection of all maximal ideals of L(Lif Lhas no maximal ideal of L). The properties of the Jacobson radicals of finite-dimensional Lie algebras have been investigated by Marshall [6] and he has shown the following
THEoREM O.1. Ifafinite-dimensional Lie algebra L has a Levi decomposition L == S + a(L), then the lacobson radical of L equals to [L, a(L)], where u(L) is the solvable radical of L.
For the infinite-dimensional case, Kamiya [2] shows
THEoREM O.2. if a Lie algebra L generated by finite-dimensional local subideals of L, then the Jacobson radical of L equals to [L, o(L)], where a(L) is the maximal locally solvable ideal of L.
These results are not ture for the general infinite-dimensional case, even for the locally finite Lie algebras. This will be seen in g4. Taking a look at the Lie algebras given in g4 it seems to be diMcult to find the characterization of the Jacobson radicals of infinite-dimensional Lie algebras by the well-known radicals. In this paper, to investigate the Jacobson radical of Lie algebras, we study the Lie algebras whose Jacobson radical is zero.
In g2 we will prove a local property that if H is an ascendant subalgebra of a Lie algebra Lthen the Jacobson radical of H is contained in that of L. This goes as well for a serial subalgebra H of a locally finite Lie algebra L.
The main result of g3 is that a locally finite Lie algebra L with null Jacobson radical is a direct sum of a semisimple ideal of L, whose Jacobson radical is zero, and the center of L.
In g4 we give the two examples of Lie algebras. These Lie algebras tell us that some results about the Jacobson radical of finite-dimensional Lie algebras are not true in the infinite-dimensional Lie algebras.
1. Preliminaries
Throughout this paper we always consider not necessarily finite dimensional Lie algebras over a field of characterstic zero unless otherwise specified. Notation and terminology are mainly based on Amayo and Stewart [1]. In particular "Åq", "si",
"asc" , "lsi" and "ser" denote the relations "ideal", "subideal", "ascendant subalgebra",
"local subideal" and "serial subalgebra" respectively. For example, a subalgebra H of a Lie algebra Lis ascendant, this is denoted by H asc L, if there is an ordnal o and a series of subalgebras {L.}.-.. such that
Lo=H, La=L,
L. Åqq L.+i for all ctÅqa,
L. =U.ÅqzL. for all limit ordinals As: o.
If a is finite then H is a subideal of L.
Let Lbe a Lie algera and Ha subalgebra of L. We denote the center ofLby 4(L) and CL(H)={xELI[H, x] =O}. Triangular bracketes Åq År denote the subalgebra generated by their contents. We also denote by adLx(y) =[y, x] for any x, yEL. Let S be a non-empty subset of L. The ideal closure SL is defined by
co sL - 2 [S, .L]
n=O
where [S,.L] is inductively defined as follow: [S,oL] = S, [S, iL] = [S, L], [S, .+iL]
= [[S,.L], L]. This is the smallest ideal of L containing S.
Let K be a Lie algebra and I a subalgebra of the derivation algebra Der(K) of K. Consider the direct sum
. L=K+I
of vectorspaces K and I. Then Lis a Lie algebra with the product: [h+i,k+j]
= [h, k] + i(k) - j(h) + [i, J'] (h, kE K, i, jEI). This Lie algebra L is called the split extension of K by I.
Let L be a Lie algebra. If every finite subset of L is contained in a finite- dimensional subalgebra (resp. a solvable subalgebra) of L then L is called locally finite (resp. Iocally solvable). For a locally finite Lie algebra L, a(L) is the maximal locally solvable ideal of Land Lis said to be the semisimple if a(L) = O. Then we have the following
LEMMA 1.1 ([1 ; Theorem 13.3.7]). Let L be a locally .finite Lie algebra. lfH ser L then o(H) g a(L).
Since a locally finite Lie algebra satisfying the minimal condition for two step subideals is finite-dimensional and solvable ([1; Corollary 8.5.5]), the following lemma
Jacobson Radical 25
is immediate.
LEMMA 1.2. The locally solvable simple Lie algebra is 1-dimensional.
We always put Lca = A.co=i L".
LEMMA 1.3 ([1; Lemmas 1.3.2,1.3.4,13.2.3]). (1) ifH si L or H lsi L then Hto Åq1 L.
(2) lf H is a .finite-dimensional ascendant subalgebra of L then Htu Åq L.
For a Lie algebra L let A(L), B(L) be the subalgebra generated by all finite- dimensional ascendant subalgberas of L, the subalgebra generated by all finite- dimensional local subideals of L respectively. We set
F.(L) =: {H asc LIH is finite-dimensional}, Fb(L) = {H lsi LlH is finite-dimensional}.
It is known that the class of the finie-dimensional Lie algebras is ascendantly coalescent and lsi-coalescent ([1; Theorems 3.2.5, Corollary 13.2.2]). In other words if H, KE F.(L) (resp. Fb(L)) then ÅqH, KÅrEF.(L)(resp. Fb(L)). Therefore we can write
A(L)=UHEFa(L)H, B(L)=UHEFb(L)H' For these radicals of L, Kubo [4] shows
LEMMA 1.4. lfL is a locally .finite Lie algebra then A(L) and B(L) are ideals ofL.
2. Local properties of L with J(L)=O
For a Lie algebra L we denote by J(L) the Jacobson radical of L.
LEMMA 2.1. (1) (Levi6 [5]) A simle Lie algebra can no non-trivial ascendant subalgebra.
(2) (Stewart [8]) A locally .tinite simple Lie algebra can no non-trivial serial subalgebra.
We state the following key lemma.
LEMMA 2.2. (1) if H is an ascendant subalgebra of a Lie algebra L then J(H) g J(L).
(2) lf L is locally .finite and H is a serial subalgebra of L then J(H) g J(L).
PRooF. Let H be an ascendant sualgebra (resp. a serial subalgebra in the case that Lis locally finite) ofLand M a maximal ideal of L. For a homomorphismfof Lit is obvious that if H asc Lthen f(H) asc f(L). If Lis locally finite and H ser Lthen f(H) serf(L) by [1 ; Proposition 13.2.4]. Hence (H + M)/M asc L/M (resp. (H + M)/M ser
L/M). Therefore HgM or H+M=L by Lemma 2.1. If H+M=L then HnM is
amaximal ideal of H. This shows our assertion. D
We shall see the structure of finite-dimensional ascendant (serial) subalgebras of a Lie algebra with null Jacobson radical.
THEoREM 2.3. if L is a Lie algebra with J(L) == O then every .f7nite-dimensional ascendant subalgebra H of L has a unique Levi decomposition
H- H2 O g(H)
where H2 is a semisimple ideal of L. Moreover L= H2 (D CL(H2).
PRooF. By Theorem O.1 and Lemma 2.1 we immediately have [a(H), H] - J(H) g J(L) - O.
Hence 4(H) == 6(H). This shows that H has a unique Levi factor H2. By Lemma 1.3 we have H2 = Hto Åq L. Let xeL. Since H2 is a finite-dimensional semisimple ideal of L, adLxlH2 is an inner derivation of H2. Hence there exists zGH2 such that adLxlH2
=adLzlH2. Therefore x-zEC.(H2) and L=H2eC.(H2). D
THEoREM 2.4. lfL is a Lie algebra with J(L) = O then everyfinite-dimensional local subideal H of L has a unique Levi decomposition
H == Hco (D o(H)
where Hto is a semisimple ideal ofL. Moreover L=Hco (D CL(Hco).
PRooF. Since H is finite-dimensional and J(L) = O, we can choose maximal ideals M,,..., M. of Lsuch that Hn(Min•••nM,) = O and Min•••nMi-inMi+in•••nM, g Mi for i= 1, ..., r. We write
L/A:•=, M, - ZO ((D e•z,S,)
where Z = C(L/A:•=i Mi) and the Si's are non-abelian simple. Considered H as a local subideal of Z O (e e•=, Si), we have Hco Åq (!) e•=i Si = (Z e (e e•=iSi))to by Lemma 1.3. Hence Hto is semisimple. Let S be a Levi factor of H. Then S == Sco g Hto and so
Hco=S. The last assertion is similarly proved as in Theorem 2.3. D
We shall describe the structures of the radicals A(L), B(L) of L given in g1 when J(L) = O, as follow.
THEoREM 2.5. lfL is a Lie algebra with J(L) ==O then A(L) = ({DAEASA Åq{D 4(A(L))• B(L) = ([IDaEASz ({D c(B(L))
where {SAIZEA} is the set of all .finite-dimensional non-abelian simple ideals of L.
PRooF. We already saw in g1 that A(L) = UH.F.(L)H and B(L) == UHEF,(L)H• We
Jacobson Radical 27
write
SA = UHEFa(L)H2, ZA = UHEFa(L) O(H), SB=UHeFb(L)Hto, ZB=UHeFb(L)a(H)'
Then by Theorems 2.3, 2.4 SA and SB are direct sums of finite-dimensional non-abelian simple ideals of L Hence SA == SB = ez.AS2 where Sz runs over all finite-dimensional non-abelian simple ideals of L.
Let H, KEF.(L) (resp. F,(L)). Since H si ÅqH, KÅrEF.(L) (resp. F,(L)), a(H)g a(ÅqH, KÅr) by Lemma 1.1. Hence [o(H), K] g [e(ÅqH, KÅr), ÅqH, KÅr] g o(ÅqH, KÅr) E ZA(resp. ZB). Therefore ZA (resp. Z.) is an ideal of A(L) (resp. B(L)).
Obviously A(L) = S. O ZA and B(L) = SB e ZB. Take any element xEZA and let xEo(H) for some HEF.(L). For any KEF.(L) we have [K, x] g [ÅqH, KÅr, a(H)] g [ÅqH, KÅr, o(ÅqH, KÅr)] =O by Theorem 2.3. This implies that Z. g 4(A(L)). Therefore g(A(L)) = Z. + S.n4(A(L)) = ZA. By the definition of Z. and the fact that Z. Åqa B(L),
we have Z. E a(B(L)). Therefo re a(B (L)) = Z. + S. na(B(L)) = ZB. D
3. Locally finite Lie algebras with null Jacobson radical
Let Lbe a finite-dimensional Lie algebra. By Theorem O.1 we can easily derive that J(L) == O if and only if L=Se4(L) where S is a semisimple ideal of L with J(S)
== O. In this section we will extend this results to locally finite Lie algebras in the following theorem 3.3. 0f couse for a finite-dimensional Lie algebras we can drop the condition that J(S) = O. But we can not drop it for locally finite Lie algebras, because there is a locally finite semisimple Lie algebra Lwith J(L) l O. Such a Lie algebra L will be given in the next section.
LEMMA 3.1. Let L be a locally finite Lie algebra. if a(L)nL2 =O then L has a Levi decomposition S (li) 4(L) where S is a semisimple ideal of L.
.
PRooF. Let S be a subspace of Lsuch that L=S+ o(L) (the direct sum of vector spaces) and L2 gS. Then S is a semisimple ideal of L. Since [a(L), L]nL2= O, we
LEMMA 3.2. ifL=ea.ALz then J(L)= ea.AJ(LA)•
PRooF. It follows from Lemma 2.2 that ez.AJ(L2) gJ(L). If Mz is a maximal ideal of LA then M, e((D,;zL,) is a maximal ideal of L. Hence J(L) g nz.A(J(La) (ID
THEoREM 3.3. LetL be a locallyfinite Lie algebra. Then 1(L) =O ofanfonly ofL has a Levi decomposition L= Se 4(L) where S is a semisimple ideal ofL with 1(S) = O.
PRooF. Let M be a maximal ideal of L. Assume that o(L)9M. Since L=M
+o(L), a
,,.,.,,..tL,)/,(,O(,L,)RiilZ3IS.,10CC','g,,S,O.:v,ab,',e.a."/d.strmgig•.,H.e,nge.o.(L)i(,a,.kLl.,ciK,,t,ii,a{
o(L) n L2 g J(L).
If J(L) = O then by Lemma 3.1 L has a Levi decomposition L= S Ot 4(L) where S is a semisimple ideal of L. By Lemma 2.2 J(S)=O.
The other implication is obvious by Lemma 3.2. Z
CoRoLLARy 3.5. Let L be a locally finite Lie algebra with J(L) == O. Then B(L) - (e,..S,) e 4(L)
where {SzlZEA} is the set ofallfinite-dimensional non-abelian simple ideals ofL. Hence A(L) - B(L).
PRooF. By Theorem 2.5 we write B(L) = (e,..S,)ea(B(L)). Since B(L) ÅqaLby Lemma 1.4, a(B(L))go(L). We have o(L)=4(L) by Theorem 3.3. Hence
a(B(L)) g 4(L) g B(L) and so a(B(L)) == 4(L).
Since SA g A(L) (ZEA) and g(L) g A(L), we have B(L) g A(L). Hence B(L) = A(L).
z
4. Examples
Example 1. Let Vbe a vector space of infinite dimension over a field of charactersitc zero. Let S the set of all linear transformations of V, regarded as a Lie algebra under the usual Lie multiplication [s, t] = st - ts (s, tGS). Let F be the set of elements ofS of finite rank and A the set of elements of F of trace zero (in the sense of [7;g4]). It is shown in [7] that A is infinite-dimensional simple. It is easy to see that A = F2 and F is locally finite. Moreover the• only ideals ofF are O, A and F. Hence a(F)=O and J(L) - A.
PRoposiTioN 4.1. There is a locally finite Lie algebra L such that L is semisimple and J(L) iL O.
Example 2. We slightly change the construction of the Lie algebra given in Kubo [3].
For any pofiitive integer i, let S, be the 3-dimensional split simple Lie algebra over k of
EY,rZC5el iSlll'C,:?.rO{ l;",?, bljllSi=S {.'`l.=K",,let} i2S,M,",Lt,'9.'IC,g.ti8"S [X`' 't] = ht' [Xt' hi] = 2Xt'
co oo
x == Z ad xi, h= Z ad h,
i=1 i--1
Jacobson Radical 29
of K. Then [x, h] =Xad [xi, hi] =X2ad xi == 2x. Hence Åqx, hÅr = kx + kh is a subalgebra of Der(K). Consider the split extension
L=K+ Åqx, hÅr
of K by Åqx, hÅr.
LEMMA 4.2. Every nonzero ideal of L contains Si for some i.
PRooF. Let H be a non-zero ideal of L. We assume that H gK. We can take a
non-zero element w of H such that w= 21=, vi + ctx + fih (viESi, ct,6Ek) where ct 7E O or 6 7E O. Since [w, S.+i] = [ctx.+i+6h.+i, S.+i] 7E O, this is a non-zero subspace of Sn+1'
Hence
Sn+1 == [W, S.+1]S""gH. []
THEoREM 4.3. (1) L is locally finite and semisimple.
(2) L is not a direct sum of non-abelian simple ideals of L.
(3) J(L) - O.
PRooF. The semisimplicity ofLfollows from Lemma 4.2. Since L2 #L, we have
the assertion (2).
(3): Since S. is finite-dimensional and simple, L/CL(S.)=Der(S.);S.. Hence CL(S.) is a maximal ideal of L. Put H = A.co= i CL(S.). If H lO then Si gH for some i
by Lemma 4.2, a contradiction. Hence H=O and J(L)gH=O. D
We describe all idels of L as follow.
PRoposmoN 4.4. An ideal H ofL is O, L, K, K+ÅqxÅr, or of the forms H == (e,.pS,) + k(x - 2 x,) + k6(h - Z h,)
qEQ qEQ
where P is a non-empty subset of N, 9 is a non-empty finite subset of N with 2nP = ip and 6ek.
PRooF. Let H be an ideal which is not one of the first four types listed above. Let P = {p ENI [S., H] : O} and e == {qENI [S,, H] = O}. By Lemma 4.2 P ; ip. If P = N then K g H, contradiction. Hence 9 # ip. Obviously H == e,.p S,
+ (e,.QS, + Åqx, hÅr)nH.
Any non-zero element w of (e,.QS, + Åqx, hÅr)nHXe,.QS, can be of the form w == Z wt + ctx + 6h
tET
(ct, 6Ek, w,eS. Tg e) where ct 7! O or 6 iL O. We may assume that w, iE O for tE T, and
choose w in such a way that 1Tl is as small as possible. Write w, = atxt + btyt + ctht (a,, bt, ctEk). By [x,, w] = [h,, w] =O we have a, = - ct, b, == O, c, = -6 and so
w - ct(x - 2 x,) + 6(h - 2 ht)•
tET tET
Assume that e iE T Then [x,, w] = 26x, and [h,, w] = - 2ctx, for qE 9XT Hence S, g H, a contradiction. Therefore T=e and 9 is a finite set. Since [w, x] ==
-26(x-Xt.Tx,)EH,His of the form required. D
Let U be a finite-dimensional Lie algebra and Vits ideal. If J(U) = O then J(U/V)
=O by Theorem O.1. But this is not true for our Lie alebra L, Because J(L) =O but J(L/K) - ÅqxÅr + K.
THEoREM 4.5. There is a locally .finite Lie algebra L and its ideal K such that J(L)
-O but J(L/K)lO.
References
[1] R. K. Amayo and I. N. Stewart, Infinite-dimensional Lie algebras, Noordhoff, Leyden, 1974.
[2] N. Kamiya, On the Jacobson radicals of infinite-dimensional Lie algebras, Hiroshima Math. J. 9 (1979), 37-40.
[3] F. Kubo, An infinite-dimensional semisimple Lie algebras, Hiroshima Math. J. 12 (1982), 607-609.
[4] F. Kubo, Notes on invariant radicals of locally finite derivations, Bull. Kyushu Inst. Tech. (Math.
Natur. Sci.) 34 (1987), 5-8.
[ 5 ] E. I. Levi6, On simple and strictly simple rings, Latvijas PSR Zinat4u Adad. Vestis Fiz Tehn. Zinat4u Ser. 6 (1965), 53-58.
[6] E. I. Marshall, The Frattini subalgebras ofa Lie algebra, J. London Math. Soc. 42 (1967), 416-422.
[7] I. N. Stewart, The minimal condition for subideals of Lie algebras, Math. Z 111 (1969), 301-310.
[8] I. N. Stewart, Subideals and serial subalgebras of Lie algebras, Hirohsima Math. J. 11 (1981), 493- 498.
Department of Mathematics Kyushu Institute of Technology
