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

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