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General Representation Theory of Jordan Algebras Author(S): N General Representation Theory of Jordan Algebras Author(s): N. Jacobson Source: Transactions of the American Mathematical Society, Vol. 70, No. 3 (May, 1951), pp. 509-530 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1990612 Accessed: 25/09/2010 15:49 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. 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]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society. http://www.jstor.org GENERAL REPRESENTATION THEORY OF JORDAN ALGEBRAS BY N. JACOBSON The theoryof Jordanalgebras has originatedin the study of subspaces of an associative algebra that are closed relative to the composition ab = a X b +b Xa wherethe X denotes the associative product. Such systemsare called special Jordan algebras. It is well known that the composition ab satisfies the conditions (0.1) ab = ba, (a2b)a = a2(ba). This has led to the definitionof an (abstract) Jordanalgebra as a (nonassocia- tive) algebra whose multiplicationsatisfies the above conditions. It is an open question as to how extensive is the subclass of special Jordan algebras in the class of Jordanalgebras. However, it is known that there exist Jordan algebras which are not special. If 2 is a special Jordan algebra, then it is natural to consider the linear mappings a-* Ua of 2I into linear transformationsUa such that (0.2) Uab = UaUb + UbUa, forthese mappingsare just the homomorphismsof 2?into special Jordanalge- bras of linear transformations.We shall now call such mappingsspecial repre- sentationsof W.Special representationshave been consideredpreviously by F. D. Jacobson and the presentauthor and complete resultshave been obtained for finite-dimensionalsemi-simple algebras of characteristic0('). There is anothermapping of 2 into linear transformationswhich is funda- mental in the structuretheory of Jordan algebras, namely, the regularmap- ping a-*Ra whereRa is the multiplicationx-*xa =ax. It is known that the Ra satisfythe followingfunctional equations: (0.3) [RaRbc] + [RbRac] + [RcRab] 0 where,as usual, [AB] denotes AB -BA and (0.4) RaRbRc + RcRbRa + R(ac)b = RaRbc + RbRac + RcRab(2). Presented as an invited address in the Algebra Conferenceof the International Congress of Mathematicians, September 2, 1950; received by the editors August 17, 1950. (1) [7 ] A (special) representationin the sense of [7 ] as a linear mapping a-+ Va, Va a linear transformation,such that Vab = (Va Vb+ VbVa)/2. If Va satisfiesthis condition, then Ua = Va/2 satisfies(0.1). The change to (0.1) is made so that special representationswill be representations in the sense definedbelow. Numbers in brackets referto the bibliography at the end of the paper. (2) Cf. [1, p. 549]. 509 510 N. JACOBSON [May As a generalization of this situation we definea (general) representationof aI to be a linear mapping a->Sa of 2? into linear transformationsSa which satisfythe functionalequation forthe R's. It is noteworthythat any special representationis a representation.Thus, the theory of representationsen- compasses both aspects of the theoryof Jordan algebras that we have men- tioned above. If 2fis a subalgebra of a largeralgebra S8, then it is clear that the correspondencea-+Ra, where Ra now denotes the multiplicationin e3 by aGC, is a representationof W. It is clear frornthis remarkthat the theoryof representationswill be a basic tool in the study of subalgebras of Jordan algebras. The main objective of the present paper is the development of the gen- eral representationtheory of Jordan algebras. The only application which we shall note explicitlyhere is to the study of semi-simplesubalgebras of an arbitraryfinite-dimensional Jordan algebra of characteristic0. In this con- nectionwe obtain the perfectanalogues of the resultsof Malcev and of Har- ish-Chandraon the theoryof the Levi decompositionof a Lie algebra [12; 5]. An important incidental tool in our discussion is another type of ab- stract systemcalled a Lie triplesystem. A special Lie triplesystem is defined to be a subspace of an associative algebra which is closed under the ternary composition [[ab]c]. It is easy to see that any special Jordan algebra is a special Lie triplesystem. Contraryto the situation which obtains forspecial Jordan algebras, it is possible (as will be shown below) to give a perfect axiomatic descriptionof special Lie triple systems. The correspondingab- stract systems-(abstract) Lie triplesystems-play an importantrole also in the theoryof abstract Jordan algebras. Thus, we can show that any Jordan algebra 2{ is a Lie triple system relative to the composition [abc] (bc)a -b(ca). Any representationof ?1 is a representationof the associator Lie triplesystem W. On the other hand, any Lie triple system can be imbedded in a Lie algebra in such a way that a representationof the Lie triplesystem can be extended to one of the Lie algebra. This observationwill enable us to apply the well developed theoryof Lie algebras to the presentproblem. The incidentalresults which we obtain on Lie triplesystems may also be of some intrinsicinterest, since these systemsconstitute a generalizationof Lie alge- bras as well as of Jordanalgebras. I. GENERAL THEORY 1. Jordan algebras and Lie triple systems. A (nonassociative) algebra 2 over a field 1 is called an (abstract) Jordan algebra if the multiplication compositionsatisfies the followingidentities: (1. 1) ab = ba, (1.2) (a2b)a = a2(ba). We writeF(a) = (a2b)a-a2(ba) and formthe second difference 1951] GENERAL REPRESENTATION THEORY FOR JORDAN ALGEBRAS 511 2F = F(x + y + z) - F(x + y) - F(y + z) - F(x + z) + F(x) + F(y) + F(z). If the characteristicis not equal to 2, A 2F=O gives (1.3) xybz+ yzbx+ zxby = (xy)(bz) + (yz)(bx) + (zx)(by), where we have abbreviated ((ala2)a3)an) to aja2 * * * an. Conversely, if the characteristicof the base field is not equal to 3, then (1.3) implies (1.2). From now on we shall assume that the characteristicis not equal to 2 or 3. If we denote the mapping x-*xa (=ax) by Ra and change the notation slightly,then from(1.3) we obtain the relations (1.4) [RaRbc] + [RbRca] + [RcRab] = 0, (1.5) RaRbRc + RcRbRa + R(ac)b = RaRbc + RbRca + RcRab. We remark that either (1.4) or (1.5) and the commutative law imply (1.3). If we interchangea and b in (1.5) and subtract, we obtain the important relation (1.6) [[RaRb]Rc] = RA(b,c,a), where A (b, c, a) is the associator (bc)a-b(ca). We definenext an (abstract) Lie triplesystem. This is a vector space over a field 1 in which a ternarycomposition [abc] is definedwhich is trilinear and satisfies: (1.7) [aab] = 0, (1.8) [abc] + [bca] + [cab] = 0, (1. 9) [[abc]de] + [[bad]ce] + [ba[cde]] + [cd[abe]] = 0, (1. 10) [[abc]de] + [[bad]ce] + [[dcb]ae] + [[cda]be] = 0, (1.11) [ [[abc]de]fg] + [[[bae]df]eg] + [ [ [bad]ce]fg]+ [[[abd]cf]eg] + Q + R = 0, where Q and R are obtained fromthe fourprevious termsin (1.11) by cyclic permutationof the pairs (a, b), (c, d), (e, f). If V is a Lie algebra and we defined [abc]= [[ab]c] in terms of the Lie composition [ab], then 3 is a Lie triplesystem relative to [abc] [10, p. 152]. In particular any associative algebra is a Lie triple system relative to [abc] = [[ab]c] where [ab ] = ab-ba. We shall now show that any Jordan algebra is a Lie triple system rela- tive to the composition [abc]=-A(b, c, a). We assume firstthat W has an identity.In this case the correspondencea->Ra is 1-1. By (1.6), R[abc] = [RaRbRc] - [[RaRb]Rc]. 512 N. JACOBSON [Mlay Since [RaRbRc] satisfies (1.7)-(1.11) the same holds for [abc]. Hence our assertionis proved. If 2?does not have an identity,then we can adjoin one in the usual fashion.The result is a Jordan algebra. We can now use the Ra in the extended algebra to prove that 21is a Lie triplesystem relative to [abc]. We shall call this Lie triplesystem the associatorsystem of W. 2. Definitionand elementaryproperties of representations. DEFINITION 2.1. A linear mapping a->Sa of a Jordan algebra 2? into the algebra of linear transformationsof a vector space 91 over 41 is called a representationif (2.1) [SaSbc] + [SbSca] + [ScSab] =, (2.2) SaSbSc + ScSbSa + S(ac)b = SaSbc + SbSca + ScSab- This concept is equivalent to that of a module which has been intro- duced by Eilenberg [4, p. 133]. We definethe latter in the followingdefini- tion. DEFINITION 2.2. A Jordanmodule is a systemconsisting of a vector space 9), a Jordanalgebra A1,and two compositionsxa, ax forx in 91, a in !f1which are bilinear and satisfy (2.3) ax = xa, (2.4) (xa)(bc) + (xb)(ca) + (xc)(ab) = (x(bc))a + (x(ca))b + (x(ab))c, (2.5) xabc + xcba + acbx = (xa)(bc) + (xb)(ca) + (xc)(ab).
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