Issa, A N

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Available at: http://www.ictp.trieste.it/~pub off IC/98/49United Nations Educational Scientic and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSA CLASSIFICATION OF TWO-DIMENSIONALHYPOREDUCTIVE TRIPLE ALGEBRASA. Nourou Issa1Institut de Mathemathiques et de Sciences Physiques,Universite Nationale du Benin, B.P. 613 Porto-Novo, BeninandThe Abdus Salam International Centre for Theoretical Physics, Trieste Italy.AbstractTwo-dimensional hyporeductive triple algebras (h.t.a) are investigated. Using theK. Yamaguti's approach for the classication of two-dimensional Lie triple systems (L.t.s), aclassication of two-dimensional h.t.a is suggested.MIRAMARE { TRIESTEMay 19981Regular Associate of the ICTP.Fax: (229)2125251 IntroductionHyporeductive algebras were introduced (see [5],[6]) as an innitesimal tool for the study ofsmooth hyporeductive loops which are a generalization both of smooth Bol loops and smoothreductive loops (i.e. smooth A-loops with monoalternative property [7]). By way of fact itis shown that the fundamental vector elds of any smooth hyporeductive loop constitute analgebra called a hyporeductive algebra of vector elds. Further (see [2],[3]) this notion hasbeen extended to the one of abstract hyporeductive triple algebras (h.t.a for brevity) meaninga nite-dimensional linear space with two binary and one ternary operations satisfying somespecic identities. It turns out that hyporeductive algebras generalize Bol algebras and Lietriple algebras (see [7],[9] about Bol and Lie triple algebras).In this paper we consider 2-dimensional h.t.a and the prospecting of explicit expressions ofoperations for such algebras led us to their classication. In section 2 some specications ofhyporeductive algebras are given and the classication theorem is stated. Section 3 is devotedto its proof (this proof gives the classication process). As a consequence it is pointed out thatthere is no proper 2-dimensional Lie triple algebra. We conclude with some remarks in section 4.2 Background and results.Hyporeductive algebras were originally introduced ([5],[6]) as algebras of vector elds on asmooth nite-dimensional manifold, satisfying a specic condition. More exactly it was giventhe followingDenition 1 [5],[6]. A linear space V of vector elds on an n-dimensional manifold with asingled out point e, satisfying[X; [Y; Z]] = [X; a(Y; Z)]+ r(X ; Y; Z) (1)is called a hyporeductive algebra of vector elds with determining operations a and r, if dimfX(e) :X 2 V g = n.Obviously a(Y; Z) is a bilinear operation and r(X ; Y; Z) a trilinear one on V and a(Y; Y ) = 0,r(X ; Y; Y ) = 0. The relation (1) we called (see [2],[3]) the hyporeductive condition for alge-bras of vector elds. Considering a hyporeductive algebra as a tangent algebra at the iden-tity e of a smooth hyporeductive loop it is shown ([5]) that a hyporeductive algebra maybe viewed as an algebra with two binary operations a(X; Y ); Te(X; Y ) = [X; Y ](e) and oneternary operation r(Z;X; Y ) and then working out the Jacobi identities in the correspond-ing enveloping Lie algebra, one can get the whole system of identities linking the operationsa; Te; r. A similar construction is carried out in [2],[3], where instead of Te(X; Y ) the operationb(X; Y ) = [X; Y ](e) a(X; Y ) is introduced (this is made in connection with a more suitable dif-ferential geometric interpretation of a hyporeductive algebra of vector elds and then the systemof identities above mentioned constitutes the integratibility criteria of the structure equationsof the anely connected smooth manifold associated with a local smooth hyporeductive loop).This led us to introduce the notion of (abstract) hyporeductive triple algebras (h.t.a):2Denition 2 [2],[3]. Let V be a nite-dimensional linear space. Assume that on V are denedtwo binary anticommutative operations (:); () and one ternary operation <  ; ;  > skew-symmetric with respect to the two last variables. We say that the algebra (V ; :; ; <  ; ;  >)is an abstract h.t.a if, for any ; ; ; ; ;  in V the following identities hold: f  : ( : )  < ; ;  > g = 0 (2) f   ( : ) g = 0 (3) f < ; ;  :  > g = 0 (4): < ; ;  >  : < ; ;  > + < :; ;  >=<   ; ;  >   <   ; ;  >+ < ; ;  >   < ; ;  >+(  )  (  ) + (  ):(  ) (5):(: < ; ;  >  : < ; ;  > + < :; ;  >)+ << ; ;  >; ;  >   << ; ;  >; ;  >+ < ; ; < ; ;  >>   < ; ;< ; ;  >>= 0 (6)  (: < ; ;  >  : < ; ;  > + < :; ;  >) = 0 (7)< ;; : < ; ;  >  : < ; ;  > + < :; ;  >>= 0; (8)where  denotes the cyclic sum with respect to ; ; .The study of h.t.a is more handy if they are given in terms of identities as in the denitionabove. For instance, we notice that if in (2)-(8) we set : = 0 for any ;  of V then we get thedetermining identities of Bol algebras. On the other hand, setting    = 0, we get Lie triplealgebras (i.e. generalized Lie triple systems) and if, moreover, we put : = 0 then we obtainLie triple systems (L.t.s).The question naturally arises whether there exist proper abstract h.t.a. The answer to thisproblem is easier to seek among low-dimensional h.t.a because of the specic properties of op-erations (:); () and <  ; ;  >. Thus we are led to the study of two-dimensional h.t.a, thatis to nd the explicit expressions for their determining operations. Our investigations led us tothe following classication theorem for such h.t.a.Theorem. There exist proper 2-dimensional h.t.a. Moreover any 2-dimensional h.t.a is iso-morphic to one of the algebras described in the table below:3type x1 x2x1:x2< x1; x1; x2> < x2; x1; x2> observationsI x1+ x20 x2x1II x1+ x20 (x1+ x2) (x1  x2)  6= 0III x1x1+ x20 x1+ x2 6= 0;  6= 0, 6= 0;  6= 0,    6= 0IV x1x20 x1+ x2 6= 0;  6= 0, 6= 0;  6= 0V  x1x10 (x1+ x2)  6= 0;  6= 0VI x2x1+ x2x1+ x20  6= 0;  6= 0, 6= 0;  6= 0    6= 0VII x2x1x1+ x20  6= 0;  6= 0, 6= 0;  6= 0VIII  x2x2(x1+ x2) 0  6= 0;  6= 03 Proof.First we shall prove the following two lemmas.Lemma 1. If fx1; x2g is a basis of a 2-dimensional h.t.a V, then the determining identities(2)-(8) of abstract h.t.a has the following type:J(x1; x2)   < x1:x2; x1; x2>+x1: < x2; x1; x2>  x2: < x1; x1; x2>= 0 (9)xi:J(x1; x2)   < xi; x1; < x2; x1; x2>>+ < xi; x2; < x1; x1; x2>>= 0 (10)xi J(x1; x2) = 0 (11)< xj; xi; J(x1; x2) = 0; (12)where J(x1; x2) = x1 < x2; x1; x2>  x2 < x1; x1; x2> and i; j = 1; 2.Proof. With respect to the basis fx1; x2g, (2),(3) and (4) are clearly satised identically. Next theleft-hand side of (5) now reads xi: < xj; x1; x2>  xj: < xi; x1; x2> + < xj:xi; x1; x2> whilethe right-hand side reads < x1 x2; xj; xi>   < xj xi; x1; x2> +xj < xi; x1; x2>  xi <xj; x1; x2> +(x1 x2)  (xj xi) + (x1 x2):(xjxi), with i; j = 1; 2. Furthermore, appealing tothe skew-symmetry of operations (:); (); (<  ; ;  >) one observes that the identity (5) getsthe form x1 < x2; x1; x2>  x2 < x1; x1; x2>=< x1:x2; x1; x2>  x1: < x2; x1; x2> +x2: <x1; x1; x2> so that we obtain (9).In view of (9), the identities (7) and (8) are straightforward transformed into (11) and (12)respectively.Finally, and again appealing to (9), we work out the identity (6) as follows: we replace ; ; ; ; by x1; x2; xk; xj; xirespectively where i; j; k = 1; 2 and then keeping in mind (9), we see that (6)gets the form4xi:(x1 < x2; x1; x2>  x2 < x1; x1; x2>)+ < xi; x2; < x1; x1; x2>>   < xi; x1,< x2; x1; x2>>= 0 that is, we get (10). }Lemma 2. If fx1; x2g is a basis of a 2-dimensional h.t.a with zero binary operations, then< x1; x1; x2>= x1+ x2; < x2; x1; x2>= x1  x2, where ; ;  are real numbers.Proof. If x1:x2= 0 and x1 x2= 0 then (9)-(12) give< xi; x1; < x2; x1; x2>>   < xi; x2; < x1; x1; x2>>= 0; (13)i = 1; 2. Now let < xs; x1; x2>= r1sx1+ r2sx2; s = 1; 2. Then, by the skew-symmetry, (13) readsr22< xi; x1; x2>  r11< xi; x2; x1>= 0 that is (r11+ r22) < xi; x1; x2>= 0. So that if r11+ r226=0,then r1i= r2i= 0 i.e. r11= r21= 0; r12= r22= 0 and this leads to a contradiction. }Remark. Lemma 2 is actually another version of lemma 5.1 in [8]. This is expected since h.t.awith zero binary operations are precisely L.t.s. Therefore the Yamaguti's classication for 2-dimensional L.t.s (see also [4], p.312) is included in 2-dimensional h.t.a of types I and II.We now set ourself about the proof of the theorem.Proof of the theorem. Since J(x1; x2) seems to be conclusive for the system (9)-(12), it wouldbe rational to work out this system according to whether J(x1; x2) is zero or not. Thus we shallconsider the two cases J(x1; x2) = 0 and J(x1; x2)6=0.A/ J(x1; x2) = 0.If we set < xs; x1; x2>= r1sx1+ r2sx2; s = 1; 2 then we get J(x1; x2) = (r11+ r22)x1 x2andJ(x1; x2) = 0 amounts to r11+ r22= 0 or x1 x2= 0.1/. Let r11+ r22= 0 and x1 x2be any.a). If r11= r22= 0, then < x1; x1; x2>= x2and < x2; x1; x2>= x1, where  and  are somereals. Therefore from (9)-(12) we draw < x1:x2; x1; x2>= 0 i.e. x1:x2= 0. Hence we get thefollowing rst set of values for < xs; x1; x2>; x1 x2and x1:x2:x1 x2= x1+ x2; < x1; x1; x2>= x2;< x2; x1; x2>= x1; x1:x2= 0 (14)(with any reals ; ; ; ).b). If r11=  r22with r116=0; r226=0, then from (9) one infers < x1:x2; x1; x2>= 0, whence x1:x2= 0so that lemma 2 gives the expressions for < xs; x1; x2>; s = 1; 2. Thus we have the followingset of operations values:x1 x2= x1+ x2; < x1; x1; x2>= (x1+ x2);< x2; x1; x2>= (x1  x2); x1:x2= 0 (15)(with any reals ; ; ;  and 6=0).2/. Now let r11+ r226=0 and x1 x2= 0.Under such assumptions the system (9)-(12) leads to the following system :< x1:x2; x1; x2>= (r11+ r22)x1:x2(16)5< xi; x1; < x2; x1; x2>>   < xi; x2; < x1; x1; x2>>= 0: (17)Whence, from the proof of lemma 2 (since (17) is (13)), we get < xi; x1; x2>= 0; i = 1; 2 andthen x1:x2= 0. Thus we obtain the trivial algebra (i.e. the algebra with zero operations) thatis included in the algebras of type (14).B/. J(x1; x2)6=0.Then r11+ r226=0 and x1 x26=0. If we set x1 x2= b1x1+ b2x2and x1:x2= a1x1+ a2x2, onends that the system (9)-(12) splits as follows:(r11+ r22)x1 x2= (a2r12  a1r22)x1+ (a1r21  a2r11)x2(18)(r11+ r22)x1:(x1 x2) = < x1; x1; < x2; x1; x2>>    < x1; x2; < x1; x1; x2>> (19)x1 (x1 x2) = 0 (20)< xj; x1; x1 x2>= 0; (21)(for i = 1), j = 1; 2, and(r11+ r22)x1 x2= (a2r12  a1r22)x1+ (a1r21  a2r11)x2(22)(r11+ r22)x2:(x1 x2) = < x2; x1; < x2; x1; x2>>    < x2; x2; < x1; x1; x2>> (23)x2 (x1 x2) = 0 (24)< xj; x2; x1 x2>= 0; (25)(for i = 2), j = 1; 2.1/. Consider the system (18)-(21). From (20) and (18) we conclude that a1r21  a2r11= 0 anda2r12  a1r226=0 (since x1 x26=0). Thus we havex1 x2= x1( 6=0); < x1; x1; x2>= 0 (from (19));< x2; x1; x2> 6=0; x1:x26=0 (from (18)): (26)Next, discussing on the coecients a1; a2; r12; r22and their compatibility with respect to (18)-(21),(26), the set of values (26) splits as follows:x1 x2= x1; < x1; x1; x2>= 0;< x2; x1; x2>= x1+ x2; x1:x2= x2; (27)(with  6=0; 6=0;  6=0;  6=0),x1 x2=  x1; < x1; x1; x2>= 0;< x2; x1; x2>= (x1+ x2); x1:x2= x1; (28)6(with 6=0; 6=0),x1 x2= x1; < x1; x1; x2>= 0;< x2; x1; x2>= x1+ x2; x1:x2= x1+ x2; (29)(with 6=0;  6=0  6=0;  6=0 and    6=0).2/. Consider now the system (22)-(25). One may notice that this system is "symmetric" to thepreceeding (18)-(21). Therefore, proceeding as in 1/, we get the following three sets of operationsvalues:x1 x2= x2; < x1; x1; x2>= x1+ x2;< x2; x1; x2>= 0; x1:x2= x1; (30)(with 6=0;  6=0;  6=0; 6=0),x1 x2=  x2; < x1; x1; x2>= (x1+ x2);< x2; x1; x2>= 0; x1:x2= x2; (31)(with  6=0; 6=0),x1 x2= x2; < x1; x1; x2>= x1+ x2;< x2; x1; x2>= 0; x1:x2= x1+ x2; (32)(with 6=0;  6=0; 6=0;  6=0 and    6=0).Hence, gathering the set of values (14), (15), (27)-(29) and (30)-(32), we get the table of thetheorem. This completes our proof. }Corollary. There is no proper 2-dimensional Lie triple algebra; more precisely the only 2-dimensional such algebras are L.t.s or the trivial algebra. Proof. As noticed in section 2, anyh.t.a fV ; ; :; <  ; ;  >g becomes a Lie triple algebra by setting x  y = 0 for all x; y in V .So that 2-dimensional Lie triple algebras must be included in algebras of types I,II. But thenx:y = 0, for all x; y in V . This means that we get either the trivial algebra or L.t.s.}4 Concluding remarks.1. The algebras of types I,II are 2-dimensional Bol algebras (proper or not). Those of typesIII-VIII are proper 2-dimensional h.t.a, and so the class of such algebras is rather wide.2. Although the survey may be conjectured tedious and lengthy, the classication of 3-dimensionalh.t.a is similar to the 2-dimensional ones. For instance, one easily checks up that any 3-dimensional h.t.a is completely determined by the given of 15 dierent binary and ternaryoperations on basis elements with respect to operations (); (:) and (<  ; ;  >). Moreover,for dimension 3 there are nontrivial identities of types (2)-(4). This latter circumstance obvi-ously makes the classication more fastidious. However it may be helpful to observe that sucha classication must include examples of 3-dimensional Bol algebras that could be drawn from [1].7Acknowledgements.This work was done within the framework of the Associateship Scheme of the Abdus SalamInternational Centre for Theoretical Physics, Trieste, Italy. Financial support from the SwedishInternational Development Cooperation Agency is acknowledged. The author would like toexpress his hearty gratitude to Professor Kiyosi Yamaguti.References[1] Fedorova V.I. On a classication of 6-dimensional Bol 3-webs. (russian). In: Webs andQuasigroups, Collect. Sci. Works, Kalinin (1984), 124-132.[2] Issa A.N. On the theory of hyporeductive algebras. (russian). In: Ryzhkov V.V. et al.(ed.),Algebraic Methods in Geometry, Collect. Sci. Works. Moskva, Izd. Ross. Univ. DruzhbyNarodov (1992), 20-25.[3] Issa A. N. Notes on the geometry of smooth hyporeductive loops. Algebras Groups Geom.,vol.12(1995), no.3, 223-246.[4] Jacobson N. Structure and Representations of Jordan Algebras. Colloquium Publ., Amer.Math. Soc., RI, vol.39(1968).[5] Sabinin L. V. On the innitesimal theory of smooth hyporeductive loops. (russian). In: Websand Quasigroups, Collect. Sci. Works, Kalinin (1990), 33-36.[6] Sabinin L. V. On smooth hyporeductive loops. Soviet Math. Dokl., vol. 42 (1991), no.6,524-526.[7] Sabinin L. V., Mikheev P. O. Quasigroups and dierential geometry. In: Chein O., SmithJ. D. H.: Quasigroups and loops, Theory and applications. Sigma series, vol. 8 (1990),Helderman Verlag Berlin, 357-430.[8] Yamaguti K. On algebras of totally geodesic spaces (Lie triple systems). J. Sci. HiroshimaUniv., ser.A, vol.21(1957), no.2, 107-113.[9] Yamaguti K. On the Lie triple system and its generalization. J. Sci. Hiroshima Univ., ser.A,vol.21(1958), no.3, 155-160.8

A classification of two-dimensional hyporeductive triple algebras

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