Derivation Algebras and Multiplication Algebras of Semi-Simple Jordan Algebras Author(S): N

Home , Center (algebra), Center (ring theory)

Annals of Mathematics

Derivation Algebras and Multiplication Algebras of Semi-Simple Jordan Algebras Author(s): N. Jacobson Reviewed work(s): Source: Annals of Mathematics, Second Series, Vol. 50, No. 4 (Oct., 1949), pp. 866-874 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1969583 . Accessed: 05/10/2012 13:40

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics.

http://www.jstor.org ANNALS OF MATHEMATICS Vol. 50, No. a, October,1949

DERIVATION ALGEBRAS AND MULTIPLICATION ALGEBRAS OF SEMI-SIMPLE JORDAN ALGEBRAS

By N. JACOBSON (ReceivedJune 9, 1948)

In thisnote we investigatethe Lie algebra of derivationsand the Lie algebra V generatedby the multiplicationsin any semi-simpleJordan algebra (with a finitebasis) over a field of characteristic0. We show that the derivation algebra Z possessesa certainideal l consistingof derivationsthat we call inner and that 3 is also a subalgebra of the Lie multiplicationalgebra S. For semi- simplealgebras we prove that i = Z. This resultis a consequenceof a general theorem(Theorem 1) on derivationsof semi-simplenon-associative algebras of characteristic0. It can be seen that anothereasy consequenceof our general theoremis the known result that the derivations of semi-simpleassociative algebras are all inner.' Our method can be applied in other cases, too (for example,alternative algebras), but we shall not discuss these here. An earliermethod of obtainingthe derivationsfor the non-exceptionalsimple Jordanalgebras has been given in a forthcomingjoint paper by F. D. Jacobson and the presentauthor.2 We compare the two resultsand we determinethe structureof the derivationalgebras. A survey of the results shows that one obtains in this way all the non-exceptionalsimple Lie algebras. In a forth- comingpaper by Chevalley and Schaferthe same problemsare solved for the exceptional simple Jordan algebra.

1. We recall the definitionof a derivationD in a (non-associative)algebra K as a lineartransformation in Wthat satisfies (1) (xy)D = x(yD) + (xD)y. Thus if we denote the rightmultiplications x -p xa by R. and we replace y by a in (1) we see that this equation is equivalent (2) [Ra XD] RaD-DRa = RaD. Similarlyif La is the leftmultiplication x ax thenD is a derivationif and onlyif (3) [La X D] = LaD . It is knownthat the set Z ofderivations is a Lie algebra oflinear transformations acting in W,that is, Z is closed under addition,under scalar multiplicationand under commutation.3 If Wis an associative algebra then

(4) RaRb = RabX LaLb = LbaX [Ra XLb] = 0

1 [7] pp. 212-215. 2 [4] 3 [7], p. 207. 866 DERIVATION ALGEBRAS OF JORDAN ALGEBRAS 867 and theserelations show that D = Rd- Ld satisfies(2) forany d. A derivation of this kind is called an inner derivation. Their totality is an ideal in the derivationLie algebra. Also if 21is an abstractLie algebrain the sense that the multiplicationsatisfies

(5) a2 = 0, (ab)c + (bc)a + (ca)b = 0 then (6) Rad = [R.; Rd] and this shows that Rd = -Ld is a derivationfor every d. Derivationsof this kind are called innerand again it is well known that theirtotality is an ideal in the derivationalgebra. We considernow the analogous facts forJordan algebras. FollowingAlbert we definesuch an algebra by the identities (7) ab = ba, (a2b)a = a2(ba). The second one of these can be linearizedto give

(8) R(.b)c = R.Rbc + RbRac + RcRb - (R.RcRb + RbRcR.)4 and this impliesthat

(9) [Ra, [Rb, Rc]] = RA(b,a,c) whereA (b, a, c) is the associator

(10) A(b, a, c) = (ba)c - b(ac). The commutativelaw gives A(b, a, c) = (ab)c - (ac)b = a[Rb, R]. Hence

(11) [Ra , [Rb, RcI]= R,[RbRcl Consequently the mapping [Rb, R], or, x -- A (b, x, c) is a derivation. It followsalso that any mapping of the formx -- >2A (b, x, c) is a derivation. Such mappingswill be called innerderivations of the Jordanalgebra. Clearly they forma subspace of the derivationalgebra. Also it is easy to verifythat if we denotex -- A (b, x, c) by Ibc then forany derivationD

(12) [Ibc, D] = I(bD)c + Ib(cD) Hence again the innerderivations form an ideal. In addition to the derivationalgebra S we shall study in this note the Lie multiplicationalgebra 2 of W. By this we mean the envelopingLie algebra,that is, the Lie algebragenerated by the multiplicationsRa . Because of (11) the set Wr= {Ra} is a Lie triplesystem of lineartransformations, that is, a subspace of

4[1] p. 549. 868 N. JACOBSON

lineartransformations that is closed relativeto the ternaryoperation [A, [B, Cl].5 It followsthat V is the totalityof mappings of the formRa + E3[Rb I R.]. Thus the subset3 of innerderivations is a Lie subalgebra of S.

2. We shall call a non-associativealgebra with a finitebasis right(left, two- sided) semi-simpleif it can be expressedas a directsum of minimalright (left, two-sided) ideals none of which are right (left,both ways) annihilatedby the algebra. Thus Wis rightsemi-simple if and onlyif the set Wrof right multiplications is completelyreducible and the set of induced mappingsin any non-zero invariant subspace (i.e., right ideal) includes non-zero mappings. Similar remarkshold for left semi-simplicityand for (two-sided) semi-simplicity. Of course,all of these conceptscoincide for commutative algebras. They coincide also forLie algebras since multiplicationis skew-symmetricin such algebras. We assume now that W is an arbitraryright (left, two-sided) semi-simple algebra that has an identityand a finitebasis over a field of characteristic0. Let e and S, respectively,be the associative and the Lie algebras of linear transformationgenerated by the right(left, right and left) multiplicationsin W. Then e and 3 are completelyreducible. It followsthat e is semi-simpleand that 3 = 2' (D L where (E is the centerand where T', the derived algebra, is semi-simple.6 Now let D be a derivationin W. The mappingD: X -- [X, D] is a derivation in the associative algebra and in the Lie algebra of all linear transformationsin the vectorspace W. By (2) ((3), (2) and (3)) the subspaces e and 2 are mapped into themselvesby D. Hence D induces derivationsin e and in S. Clearly D maps S' into itselfand it maps the center (E of2 into itself. If C e (S, C is also in the centerof A. Since the centerof e is a direct sum of separable fieldsany derivationin e maps the elementsof the centerof e into 0. Thus CD = 0. Now it is a well-knownresult of Cartan's that any derivationof a semi-simple Lie algebra 5' over a field of characteristic0 is inner.' It followsthat any derivationof a directsum 3' (D (E that maps (E into 0 is also inner. Thus there existsan elementU in V such that [R., D] = [R. X U] ([La XD] = [La) U], both conditions)hold forall a in W. Consequently (13) aD = 1RaD = 1[R., D] = 1[Ra, U] (13) -~~~~aU - (1U)a or (13') D = U - Liu. Similarlyif Wis leftsemi-simple then thereis a U in the envelopingLie algebra of the leftmultiplications such that (14) D = U-RRu.

5 Systemsof this type are consideredin [9]. 6 The resulton associative algebrasis well known. See forexample [5] p. 70. The Lie algebra resultis givenin [6] p. 878. 7 [2] p. 113 or [3] p. 689. DERIVATION ALGEBRAS OF JORDAN ALGEBRAS 869

Finally if Wis semi-simplethen thereexists a U in the Lie algebra generatedby the R. and the Lb such that (13') and (14) hold. Either one implies that D is in V. THEOREM 1. Let 2 be a right(left, two-sided) semi-simple algebra with an identitythat has a finitebasis overa fieldof characteristic0. Then if D is any derivationin 21there exists an elementU in theenveloping Lie algebraof theright (left,right and left)multiplications of 21such thatD = U - L1u (D = U -Ru, D= U).

3. Albert has definedthe radical of a Jordan algebra to be the maximal solvable ideal.8 Solvabilityis definedas for Lie algebras; 25 is solvable if the l sequence of dyadic powersof A, namely, S2 = 25S =-22, - - - leads to 0. If S is solvable it is nilpotentin the sense that thereexists a positiveinteger N such thatany productof N factorsis 0. If the base fieldhas characteristic0 then Albert has proved that the radical can be determinedby a trace condition. Let r(a) = trace Ra , then the radical 91 is the totalityof elementsz such that

(15) r(xz) = 0 forall x in 21. Also Alberthas shownthat if 9 = 0 then 21is semi-simpleand possessesan identity. Converselysuppose that 21is semi-simplein our sense. Since the radical 9 is an ideal, completereducibility of the set JR,,}implies the existenceof a com- plementaryideal e such that 2 = 9 G A. If 9 5 0 the nilpotenceyof 9 impliesthat there are elementsz F 0 in 9 such that z9 = 0. The set 3 of theseelements satisfies 391 = O and 3Q1_ 9 f e3 = 0. Hence 321 = O contrary to the definitionof semi-simplicity.Thus 9 = O. Also Albert's resultsnow show that 2 has an identity. If 21is a Jordanalgebra withan identity1, thenthe set (1) of multiplesof 1 is characteristicrelative to derivationsin the sense that it is mapped into itselfby every D in the derivationalgebra. For it is clear that iD = 0 for any D. Anothercharacteristic subspace is definedby the conditionr(y) = 0. This is clear from(2). The space &o definedby this conditionis eitherthe whole space 21or it has dimensionn - 1, n the dimensionof 21. If 21has an identityit is clear that (1); O21. Hence 21 = (1) 0 2Og. We can use the tracecondition also to provethat the radical 9 is characteristic. For if z e 9 and D is a derivationthen

r(x(zD)) = trace R,.(xD) = trace R(xZ)D-(xD)z

= trace R(XZ)D - trace R(,D)Z = 0 forall x. Hence zD e W. We considernow the theoryof derivationsfor semi-simpleJordan algebras. We prove firstthe following

I The resultson the structuretheory of Jordanalgebras that are stated here without proofsare givenin Albert'spaper [1]. 870 N. JACOBSON

THEOREM 2. Everyderivation of a semi-simpleJordan algebrawith a finite basis overa fieldof characteristic0 is inner. PROOF. We have shownthat W has an identity. Hence Theorem 1 is appli- cable. Also we knowthat V is the set of mappingsof the formRd + a [RbI Rc]. Hence D = Rd + Z[Rb, RJ]. Hence iD = d + Z(bc - cb). Since iD = 0Od = OandD = >,[Rb, Rc].

4. We shall determinenext the structureof the derivationalgebras of semi- simpleJordan algebras. It is easy to see that we have a directsum reduction to the case of simpleJordan algebras. Hence we restrictour attentionto these. The simple Jordan algebras over a field of characteristic0 fall into three "great" classes and one exceptionalclass.9 The firstgreat class consistsof the Jordanalgebras that are obtainedby usingthe multiplicationas b = 1/2(ab + ba) in simpleassociative algebras 21. We denote by 2j the Jordanalgebra that is obtained froma particularassociative algebra 2 by this process. The second great class consistsof the Jordansubalgebras (2, J) of J-symmetricelements in a simple involutorialalgebra 2 with involutionJ. The thirdgreat class is constitutedby the algebras that define Cliffordsystems. These have bases 1, u1, u2, * U**,n such that 1 is the identityand

(16) 4 = a#i 0 UiUj = 0 if i # j. Finally we have the exceptionalJordan algebras corresponding to the systemMa discoveredby Jordan,v. Neumann and Wigner. These have 27 dimensions over theircenters. The derivationalgebras of the non-exceptionalJordan algebras have been determinedin a joint paper by F. D. Jacobsonand the presentauthor using the above representationsof these algebras.10 For the firsttype we showed that any derivationhas the formx -* [x,d] xd - dx. If we denotethis mapping as Dd it is clear that Dd = 0 if and only if d is in the center. The correspondence d -* Dd is linearand D[dl ,d2] = [Ddl, Dd2]. It followsthat the derivationalgebra Z of 2 is isomorphicto 2b/ S whereWI is the Lie algebra obtainedby replacing ordinarymultiplication in 2 by commutationand G is the center. It is also easy to see that 2, = 21I G whereWI is the derivedalgebra. Hence -- 2I. We now comparethese resultswith those that can be obtainedfrom Theorem 2. Applying Theorem 2 we see that any derivation in 2s has the form x -H A(b, x, c). We can verifythat (17) A(b, x, c) = (b * x) * c-b * (x - c) = 1/4[x,[b, c]].

Hence the innerderivations x -* ZA (b, x, c) has the formx -* [x,d] notedabove. For the Jordanalgebra t (A, a) it has been proved that the derivationshave the formx -* [x, d] whered is J-skew."1 The envelopingassociative algebra of

9 [4] and [10]. 10 [4]. 11 [4]. DERIVATION ALGEBRAS OF JORDAN ALGEBRAS 871

&(W, J) is W. Hence if the derivationDd = 0, d is in the centerof K. If J is an involution of firstkind the elements of the center are symmetric.12 It followsthat if d is skew and Dd = 0 then d = 0. It followsthat the derivation algebra Z is isomorphicto the Lie algebrae(K, J) of J-skewelements (commutation as the product). By Theorem 2 and (17) we see that any derivationin S(A, J) has the form

x -* [x, Z[b, c]] where b and c are J-symmetric. Now it is clear that the commutatorof two J-symmetricelements is J-skew. Hence this checks with the above. We remarkalso that it is a consequence of these resultsthat any J-skewelement can be expressedas a sum of commutatorsof J-symmetricelements if J is of the firstkind. In a similar mannerwe see that if J is of second kind, then the derivationalgebra of S&(2t,J) is isomorphicto 5(2t,J)'. Moreover,any element of S(K, J)' can be writtenas a sum of commutatorsof J-symmetricelements. In the case of Jordanalgebras definedby (16) any derivationmaps the space spannedby the ui into itself. Thus uiD = fluss iD = 0 and the matrix(M) is J-skewwhere J is the involutiondefined by (Q) -+ (a) (t)'(a)[1 where (a) = diag {Cil, a2, aV, , an I It followsthat the derivationalgebra is isomorphicto the Lie algebra of J-skew matricesdetermined by this J. The derivationalgebra Z ofthe exceptionalJordan algebras will be determined in a forthcomingjoint paper by Chevalley and Schafer. Their resultis that if the basis fieldis algebraicallyclosed then the derivationalgebra of the (single) exceptionalJordan algebra Mg is the exceptionalsimple Lie algebra F4 of 52 dimensions. It followsthat the derivationalgebra of any exceptionalJordan algebra is of typeF4 in the sense that if its centeris extendedto its algebraic closure,then the resultingalgebra is F4 . A surveyof these results shows that withthe exceptionof a feweasily enumer- ated cases oflow dimensionalitythe derivationalgebras of simple Jordan algebras are simple Lie algebras."3 Moreover,every non exceptionalsimple Lie algebra can be obtained in this way. Finally the exceptionalLie algebra of 52 dimensions can be obtained as a derivationalgebra of a simple Jordanalgebra.

5. We shall determinenext the structureof the Lie algebras 2 forthe simple Jordanalgebras. It will be convenientto denote the Jordanalgebra as before by W. For the firsttwo classes we have 1 - 1 imbeddingsin associativealgebras, that is, we have a 1 - 1 mappinga -- Ua where Ua is in the associativealgebra such that this mappingis linear and (18) Uab = U.p Ub3 1/2(UaUb + UbUa).

12 An involutionof a simpleassociative algebra is of firstkind if it leaves the elements of the centerfixed and it is of second kindif it induces a non-trivialautomorphism in the center. 18 [8]. 872 N. JACOBSON

For the firstclass the Ua forma simple associative algebra while in the second the U. are the J-symmetricelements of a simple involutorialalgebra. In the firstcase we shall replace our originalimbedding by a more convenientone. This is obtained by formingthe direct sum of the imbeddingalgebra and an algebra anti-isomorphicto it. The elementsof the new algebra have the form U + V where multiplicationis multiplicationof componentsand where U denotesthe imageof U undera definiteanti-isomorphism between the component algebras. The mappingJ: U + V -* V + U is an involutionin the directsum, whose symmetricelements have the form U + U. The correspondence a -> Sa = Ua + UT is a 1 - 1 imbeddingof the Jordanalgebra W. Thus we now have the same situationas forthe second class, that is, we have a 1 - 1 imbeddingin an involutorialassociative algebra, now denoted as U, such that the set ofelements corresponding to 2[is the set & (U, J) of J-symmetricelements. For the sake of uniformitywe denote the imbeddingin both cases by a -+ .14 Now let A denotethe envelopingLie algebra of the elementsSa . We propose to show that V is isomorphicto A. First we shall show that the set t (U, J) is a Lie triplesystem isomorphic to the Lie triplesystem 2r of multiplicationsRa. Considerthe mappingRa -* 1/2Sa . This is 1 - 1 and linear. Moreover by (9) and (17) [Ra I [Rb, Rc]] = RA(b,a,c) -* 1/2 SA(ba c) - = 1/2((Sb Sa) * Sc Sb * (Sa *Sc)) = 1/8[Sa, [Sb , Sc]]

= [1/2Sa, [1/2Sb I 1/2Sc]].

This proves that the mappingR. -* 1/2 S. is an isomorphismfor the Lie triple system. Suppose now that Ra + Z[Rb , Rd] = 0. Then 1(Ra + 1i [Rb , Rc]) = 0 and thisgives a = 0. Hence Ra = 0, Sa = 0 and Z[Rb, Rd] = 0. This last relation impliesthat Z[Rz [Rb, Rc]] = 0 forall x. Hence Z[Sz [Sb , Sc]] = 0. Thus Z[Sb , Sc] is in the centerof U. Since the envelopingassociative algebra of the S, is U and U is semi-simplethis implies that Z[Sb, Sc] = O.'5 Thus if R. + Z[Rb , Rc] = 0 then 1/2 S. + 1/4 Z[Sb , Sc] = 0. Conversely, suppose that 1/2 S. + 1/4 Z[Sb , Sc] = 0. Then since S. is J-symmetric and [Sb6 Sc] is skew, SL = 0 and Z[Sb, Sj] = O. Since a -* S, is 1 - 1 this implies that a = 0 and hence that R. = 0. Also as before Z[Rz X[Rb, Rj]] = 0. By (9) A (b, x, c) = 0. Hence x(Z[Rb , Rj) = 0. Thus Z[Rb , Rd = 0. It is now immediatethat the mapping

(19) R. + Z[Rb, Rc] -* 1/2 S, + 1/4 Z[Sb, SC] is 1 - 1 and linear. Since commutationin V and in A are determinedby the

14 The imbeddinga -+ Sa is a "universal" imbeddingin the sense of [4]. 15 [6] Lemma 4, p. 877. DERIVATION ALGEBRAS OF JORDAN ALGEBRAS 873 ternarycommutation in W,and in S~(U, J) respectivelyit followsthat (19) is an isomorphismof S into A. It is fairlystraightforward to determinethe elementsof A. If K is in the second class and J is an involutionof firstkind then we know that any J-skew elementis a sum of commutatorsof J-symmetricelements. Hence any J-skew elementas well as any J-symmetricelement is in A. Thus A = U1 . If J is ofsecond kind (E in the secondgreat class) thenwe assertA = U' (D Go where'5o is the set ofsymmetric elements of the center. ClearlyA = ('5o. More- over since any elementof U is a sum of J-symmetricelement and a J-skewele- mentany elementof UI'is a sum of elementsof the forms[H, H'], [H, S], [S, S'] whereH, H' are J-symmetricand S, S' are J-skew. Since [H, S] is J-symmetric such an elementbelongs to A. Evidently[H, H'] is in A and as we noted before any elementof the form[S, S'] is a sum of commutatorsof elementsof S,(U, J). Hence any [S, S'] isinA. ThusA: U1 . HenceA 2 U' + Go. If A DU1 + (Yo the decompositionU = U1 0 ('5, (E the whole center,shows that A containsan elementof (E that is not symmetric. It followsthat A containsa non-zeroskew elementof the center. On the otherhand the elementsof A that are J-skeware in U1 and this contradictsu1 nf = 0. Thus A = U1 (0 (So. In a similarmanner we can treatthe case in whichW is in the firstclass. We observe firstthat a commutatorof J-skew elementsis also a commutatorof J-symmetricelements. For let U - U and V - vPbe skew. Then

[U - . V - 'V = [U. V] + [U. V] = [U + U. V + V]. It followsfrom this that A 2 U . Also A contains (Sothe totalityof elements of the formC + C, C in the centerof the firstcomponent algebra. It follows as in the precedingcase that A = U' 03 ('o . In all of the cases consideredthus farA is a directsum of a commutativeLie algebra and one or two simpleLie algebras of type A. We consider next the Lie multiplicationalgebras for the Jordan algebras definedby (16). If we use the basis (1, U1, u2, .**, un) the matrix of Rb, b = Sol + E ,hu is o0 1 * n

fOiai iOo (20) * .

O~n~en 300 As beforewe denote by J the involution(Q) (a) ()'(a)-l in the n X n matrix algebra. If we recall that the derivationsof W have as matricesrelative to (1, u1, . , un) the matrices (21) (0 ) 874 N. JACOBSON where (A) is J-skew,we see that e is isomorphicto the Lie algebra of matrices that are sums of matricesof the form(20) and of the form(21). The latterLie algebra is a direct sum of a commutativeLie algebra and the Lie algebra of matricesof the form

0 Ond

(22) G(O) I~~~~~~~~~~~

It can be verifiedthat this algebra can be characterizedas the set of K-skew matricesof (n + 1) rows whereK is the involution(t) -> (y)(t)'(y)y' where

(y) = diag {-1, al, a2, * ., an. This is a Lie algebra of type B or D. It may be noted also that by Theorem2 any matrixof the form(21) is a sum ofcommutators of matrices of the form(20). The determinationof 2 forthe exceptionalJordan algebra will be given in the paper by Chevalley and Schaferreferred to above. Here it will be shown that if the base fieldis algebraicallyclosed then 2 = (1) + E6. Where E6 is the thirdexceptional simple Lie algebra (of 78 dimensions).

YALE UNIVERSITY

BIBLIOGRAPHY [1] A. A. ALBERT, A structuretheory for Jordan algebras, Ann. ofMath. 48 (1947),546-567. [2] E. CARTAN, Th8se, Paris, 1894. [3] G. HOCHSCHILD, Semi-simplealgebras and generalizedderivations, Amer. J. Math. 64 (1942), 677-694. [4] F. D. JACOBSON AND N. JACOBSON, Structureand representationof semi-simple Jordan algebras,to appear in Trans. Amer.Math. Soc. [5] N. JACOBSON, Theoryof Rings,New York, Amer.Math. Soc. 1943. [61 , Rationalmethods in thetheory of Lie algebras,Ann. of Math. 36 (1935),875-881. [7] , Abstractderivation and Lie algebras,Trans. Amer. Math. Soc. 42 (1937), 206-224. [81 , Simple Lie algebrasover a fieldof characteristic zero, Duke Math. J. 4 (1938), 534-551. [9] , Lie and Jordantriple systems, Amer. J. Math. 71 (1949), 149-170. [101R. D. SCHAFER, The exceptionalsimple Jordan algebras, Amer. J. Math. 70 (1948), 82-94.