Bull. Kyushu Inst, Tech.
(Math. Natur. ScL) No. 41, 1994, pp 19--26
A CLASS OF DOUBLE LIE ALGEBRAS ON A LIE ALGEBRA
By
Toshiharu IKF.DA
(Received November 30, 1993)
Introduction
Recently Sanami and Kikkawa have introduced in [12] the notion of projective double Lie algebras on a Lie algebra. This notion has arisen for the determination of certain local Lie loop structures on a Lie group G. It was indeed shown by Kikkawa (see [9, Theorem 3.3], [IO, Theorem 7.3]) that the determination of the geodesic homogeneous local Lie loops in projective relation with G is reduced to that of the projective double Lie algebras on the Lie algebra of G. Succeedingly Sanami and Kikkawa [12] determined the projective double Lie algebras on odd-dimensional real simple Lie algebras. Except this fact, however, very little is known about this notion, and there was a gap in their proof. Therefore it is desirable for us to know about projective double Lie algebras on some significant Lie algebras. In this paper, including semisimple Lie algebras, we are mainly concerned with certain group-graded Lie algebras which are not necessarily finite-dimensional. In Section 2 we shall determine the projective double Lie algebras on a complete group-graded Lie algebra (Theorem 1). As a corollary those for neoclassical real semisimple Lie algebras will be determined. In Section 3 an example of a projective double Lie algebra on a Lie algebra will be presented which is different from the one gotten in Section 2.
The author would like to express his thanks to Dr. M. Sanami for his valuable comments.
1. Preliminaries
Throughout the paper Lie algebras are not necessarily finite-dimensional over an arbitrary field F of characteristic zero unless otherwise specified. Let g be a Lie algebra over F with Lie bracket [ , ] and b another Lie algebra with Lie bracket [,]b on the same underlying vector space as g. If the relation
adb(b) c Der (g)
holds, then b is called a projective double Lie algebra on g, where adb denotes the
adjoint representation of b and Der(g) the derivation algebra of q.. The source of
this notion is given in [9].
20 Toshiharu lKEDA
The centroid r(g) of q. is defined to be the set of all linear endomorphisms o such that o and ad,(X) are commutative for all Xeg. For a linear endomorphism a of g such that oEI7(g), we denote by g. the Lie algebra b with the Lie multiplication [X, Y]b =o([X, Y]). It is easy to see that g. is a projective double Lie algebra on g for any aEIi (g). In particular, g,=g,i is the most elementary example of a projective double Lie algebra on g, where I is the identity mapping of q. and pEF. We shall call thjs elementary one scalar. A linear endomorphism J of a real Lie algebra g is said to be a complex Lie structure if J2 == -I and JEI"(g). In this case, g is even-dimensional if g is finite-dimensional. It is well known that there are two distinct types of finite-dimensional real simple Lie algebras (cf. [7, Theorem 10.1 and 10.2], [3, fi4.2]). One is a real form of a complex simple Lie algebra and the centroid I-'=RL The other has a complex Lie structure J and r== RI+RJ. It will be shown in Corollary 1 that any projective double Lie algebra on a finite-dimensional real simple Lie algebra is induced by an element of L
We shall use some of the notation and terminology of [1]. For sake of convemence we state here some of them. Angular brackets Åq År denote the subalgebra generated by their contents. The left-normed products [ai,•••,a.] are defined recursively by [ai,•••,a.+i]=[[ai,•••,a.],a..i] for aicg. For a,bcg, we denote by ÅqgbÅr the smallest b-invariant subalgebra of g containing a. The centralizer C,(a) of a is defined by C,(a) = {XEq.1[X, a] == O}, and C(g) == C,(g) is the center of g.
2. A-graded Lie algebras
In this section we shall consider not necessarily finite-dimensional group-graded Lie algebras over an arbitrary field I7 of characteristic zero. Let A be an additive torsion-free group and g == (iE) g" an A-graded Lie algebra with a finite-dimensional aeA
abelian Cartan subalgebra c such that c=gO. We suppose that g has finite- djmensional irreducible c-submodules ai,•••,a. such that
(a) ai is contained in some gei (O ii! e,EA;i-- 1,•••,n), n
(b) g is generated by c+ E ai, i=1
(c) for any pair of indices i and .J' there is a sequence of indices io, ii,•••,i, such that i= io, .i -- i, and [ai., ai,,•••,ai.] 7f! O•
We shall call these c-submodules connected generators relative to c. For example, finite-dimensional split simple Lie algebras have connected generators.
LEMMA 1. Lel g=eg" be an A-graded centerless Lie algebra with .finite- aeA
dimensional abelian Cartan subalgebra c=gO and connected generators relative to
c. Suppose that o is a linear endomorphism qf g satisfbeing [o(X), Y] = [X, o(Y)] .for
A CIass of Double Lie Algebras on a Lie Algebra 21
all X, YEg. Then a is either a linear isomorphism or nulL
PRooF. Let ai,•••,a. be connected generators relative to c. We make an extension of the base field F if need be. Let K be an algebraic closed uncountable field containing F and let g=g(g)FK. We use bar convention. For any subspace a of g we denote the K-subspace a(g) FK by a. Then i is a Cartan subalgebra of g, and g has a root space decomposition g=i+ 2 gct relative to C since g is generated by i+ 2 eq. We can choose a Kd-basis {T,} of i such that nyÅëU Ker(ct) for all i, because the set of all roots d is at most enumerable. For any X.Egct and T,, if [X., T,] =O then X. = O. Therefore we can see that adg(T,) is injective on 2 gct. Let
a6A
5 =a(g) IK be the K-linear endomorphism of g extending a. Then 5 satisfies the same condition as a. Since [6(Ti), T,] == O, we have 5(T,)Ei. Thus o(c)= 5(c)cing
=: c• Now let [a;•,,•••, t,•.] be a finite-dimensional i-submodule of some gb such that b= ej,+•••+ej,lO. Then since [[th• ,•••, a;•.], Tt] =[a-j,,•••, a-,+,] and [5([al•,,•••,a-j.]),
ny] = [[ aj, ,•••, aj,], o( Ti)] c [aj, ,•••, aj.], we obtam a([aj, ,•••, aj,]) c [aJ•, ,•••, qi,]
+i and so u([aj,,•••,a,•,])c [aj,,•••,aj.] + c. In particular o(aj) c aj +c for any ]'.
We first assume a(c) = c. Then for any XGKer (o) we have [X, c] = [X, o(c)] = [o(X ), c] = O. Hence Ker (a) c c, From the assumption it foHows that a is injective. Obviously g is spanned by the finite-dimensional c-submodules of the form [aj,,•••,aj.]+c such that ej,+•••+ej,fO. As we have seen in the last paragraph o([aj,,-••,aj.])c [aj,,•••, aj,] + c. Therefore by the injectivity of a we have a([aj,,•••,aj.]
+ c) = [aj,,•••,qi.] +c Consequently a is bljective.
Next we assume 6(c)sc. Then o(T)=O for some TEcX{O}. Since g has null center, we can see [ai, T] lO for some i. We have already seen that o(ai) cc+ ai. If a(a,) gE c, then there exist Xi, YiEaA{O} and H,Ec such that a(Xi)=Hi+ Y,. Then we have [Y,, T] = [o(X,), T] == [X,, a(T)] - O. This implies that [Åq Y,CÅr, T] = O.
However, since ai is an irreducible c-submodule of g, we have ai=Åq\,"År and [a,, T] =O. This is a contradiction, whence o(ai)cc. Further if a(ai) 7i O, then [a(ai), aj] lO for some j. Since [a(ai), aj] = [ai,a(aj)] and a(aj)cc+ aj, we have aj c ai + [ai, aj] c gei + ge`'ej. This holds only if ai == aj. In this case, however, for any X,EaA {O} we have [Xi, a(X,)] == O and so [ÅqXi"År, a(Xi)] = O. Hence [a,, a(a,)]
=O, which is a contradiction. Thus a(ai) == O.
We here claim that a([ai,aj])==O for any j. For that we may suppose
[a,,aj] i7E O. If a(aj)Åëc, then there exists XjEajX{O} such that [a,,Xj] == O since [ai, o(aj)] = [a(ai), aj] = O. For any rEN and Hi,•••,H,Ec, it can be inductively seen that [ai,[Xj,Hi,•••,H,]]=O. It follows that [ai, aj] = [a,,ÅqXj"År]=O, which is a
contradiction. Hence a(aj) c c, and we can see o(aj) =O in the same way as ai. Then since [a([ai, aj]), c] = [[ai, aj], o(c)] c [[a,, a(c)], aj] + [ai, [aj, a(c)]] = O, we have
a([a,, a,•])cc. Besides [ff([a,, aj]), ak] =[[ai, aj], a(ak)]c[[ai, o(ak)], aj]+ [ai,
22 Toshiharu lKEDA
[a(ak), aj]] == O for any ak. Thus o([ai, aj])=O as claimed.
In the same way, we can see a(Eai,aj,,•••,aj.])=O for any rEN and
1gji,•••,J', s{n by induction. It follows that a(Åqa?•År)=O. Now for anyj there exist indices io, ii,•••,i,+i such that i= io, .i = i,.i and [ai, ai,,•••,ai., aj] itE O. Let pi be
the natural projection from g to Z g" and ai =pio. Then ai(aj) is a c-submodule atO
of aj. Since [ai, ai,,•••,ai., aj] 7E O and [[ai, ai,,•••,ai.], a(aj)] = [o([ai, ai,,''',ai.]), aj]
= O, we have a,(aj) = O. Then o(aj) c c, and we can obtain a(Åqa,g•År) == O in the same
way as ai. Hence we have a(c)=O since o(c)cc and [aj,ff(c)]=O for all n
1'. Consequently we have a=O since g=c+ Z ÅqaY•År. This completes the proof.
j--1
REMARK. We note that if the ljnear endomorphism a in Lemma1 has degree
zero, that is, a(g") c g" for any aEA, then aEI'(g). We note also that if c is splitting
then a =pl for some pEF.
LF.MMA 2. Let g be the same A-graded Lie algebra cts in Lemma 1. Then the
centroid I'(g) of' q. is a finite ej)ctension field qf F=FI. In particular, tf c is a split Cartan subalgebra qf g, then T(g) = FI.
PRooF. It is clear that every element o of 17= I](g) satisfies the condition of Lemma 1. Hence if OlffEr then a has an inverse mapping. Therefore r is a division ring. Let a,TEI-r. Then a and T are commutative on [g, q.] since aT[X, Y] = [a(X), T(Y)] =T[o(X), Y] == Ta[X, Y] for all X, YE q.. Moreover for any TEc and X,Ea,, we have [X,, oT(T)] = aT([Xi, T])=: Ta([X,, T])= [X,, Ta(T)]. It follows that o and T are commutative also on c. Thus I] js a field. As we have seen in the proof of Lemma 1, a(c) =O implies a=: O. Hence r is embedded in the linear endomorphism ring EndF(c) of c. Therefore the field I" is a finite extension of F.
We now suppose that cis a split Cartan subalgebra of g. Then each ai in the connected generators is one-dimensional and contained in some root space ga. We ct(aT)
choose TEc so that ct(T) 7! O. For any aEI7 we put d=a- L Then o'E]r .(T'i
ct(aT)
and aicKer(d). From Lemma 1 it follows that a=
-IEF. This completes ct(T)
the proof. '
A centerless Lie algebra g is called complete if any derivation of g is inner. The following is the main result of this section.
THEoREM 1. Let g=eg" be a complete A-graded Lie al.qebra ivith a aEA
finite-dimensional abelian Cartan subalgebra c == gO and connected generators relative to c. Then any projective double Lie algebra on g has the form g. for some oel7(g).
In particular, if c is a split Cartan subalgebra ofi g, then a is scalar.
ACIass of Double Lie Algebras ona Lie Algebra 23
PRooF. Let b be a non-abelian projective double Lie algebra on g. By the completeness of g, for any XEb there exists a unique element a(X)eg such that adb(X)= ad,(a(X)). Then o gives a Lie homomorphism from b to g. In fact, we have ad,(o([X, Y]b)) -= adb([X, Y]b)
= adb(X) adb(Y) - adb(Y) adb(X) =: ad,(a(X)) ad,(a(Y)) - ad,(6(Y)) ad,(o(X)) - ad,([a(X), a(Y)])
for all X, YEb. Since g is complete, we have a([X, Y]b) == [a(X),a(Y)] for 'all X, YEb.
Now for any X, YE .q, we have [o(X), Y] = [X, a(Y)]. From Lemma 1, it follows that the algebras b and g are isomorphic. On the other hand, since a([X, Y]b) = [a(X), a(Y)]=[u(X), Y]b for any X, YEb, we have oGI](b). It is easily seen that T(b)cr(g). Thus we have b == g.. Moreover if c is splitting, then a is scalar from Lemma 2.
CoRoLLARy 1. Let b be a prqiective double Lie algebra on a finite-dimensional real simple Lie algebra g.
(1) lf q. has no complex Lie structures, then b= g, .for some real number p.
(2) lf g has a comple.v Lie structure J, then b :q. ,.,J ,for some real numbers p and q.
PRooF. If g has a complex Lie structure J, then g can be written in the form s+ Js for some real split simple Lie algebra s. In this case, as connected generators relative to sO + JsO we can take scti + Jscti,•••,sctn + Jsan, s-ai + J.q. -ai,•••,s-atn + Js'ctn, where cti,•••,ct. are the simple roots relative to a Cartan subalgebra sO of s. Since r(g) =RI+RJ, the statement (2) follows from Theorem 1.
Suppose that g has no complex Lie structures. Then we consider g=g(EbRC Since the complex Lie algebra g is simple, we have r(g)= CI by Lemma 2. Hence
I"
