Balanced Freudenthal Kantor Triple Systems and Noncommutative Jordan Algebras

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 294 (2005) 19–40 www.elsevier.com/locate/jalgebra (−1, −1)-Balanced Freudenthal Kantor triple systems and noncommutative Jordan algebras Alberto Elduque a,∗,1, Noriaki Kamiya b,2, Susumu Okubo c,3 a Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain b Center for Mathematical Science, The University of Aizu, Aizu-Wakamatsu 965-8580, Japan c Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA Received 8 September 2004 Available online 12 October 2005 Communicated by Efim Zelmanov Abstract A noncommutative Jordan algebra of a specific type is attached to any (−1, −1)-balanced Freudenthal Kantor triple system, in such a way that the triple product in this system is determined by the binary product in the algebra. Over fields of characteristic zero, the simple noncommutative Jordan algebras of this type are classified. 2005 Elsevier Inc. All rights reserved. Keywords: Generalized Jordan triple system; Balanced Freudenthal Kantor triple system; Noncommutative Jordan algebra; Quadratic; Lie superalgebra * Corresponding author. E-mail addresses: [email protected] (A. Elduque), [email protected] (N. Kamiya), [email protected] (S. Okubo). 1 Supported by the Spanish Ministerio de Ciencia y Tecnología and FEDER (BFM 2001-3239-C03-03), the Ministerio de Educación y Ciencia and FEDER (MTM 2004–08115-C04-02), and by the Diputación General de Aragón (Grupo de Investigación de Álgebra). 2 Partially supported by a Grant-in-Aid for Science Research (No. 15540037(C)(2)) from the Japan Society for the Promotion of Science. 3 Supported in part by U.S. Department of Energy Grant No. DE-FG02-91ER40685. 0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2005.09.003 20 A. Elduque et al. / Journal of Algebra 294 (2005) 19–40 1. Introduction The well-known Tits–Kantor–Koecher construction [12,15,22] relates Jordan systems to 3-graded Lie algebras. In [14], several models of exceptional Lie algebras with a 5-grading g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 are given, based on generalized Jordan triple systems: Definition 1.1. A vector space J over a field F , endowed with a trilinear operation J × J × J → J , (x,y,z)→ xyz,issaidtobeageneralized Jordan triple system (GJTS for short) if it satisfies the identity: uv(xyz) = (uvx)yz − x(vuy)z + xy(uvz) (1.2) for any u,v,x,y,z ∈ J . A (linear) Jordan triple system is then a generalized Jordan triple system with the added constraint xyz = zyx (1.3) for any x,y,z. Unless otherwise stated, all the algebras and algebraic systems considered will be as- sumed to be defined over a ground field F of characteristic not 2, even though the results and proofs of Section 2 (respectively Section 3) are valid for systems over an arbitrary unital commutative associative ring without two torsion (respectively two and three torsion). Given two elements a,b in a GJTS J , consider the linear maps la,b,ka,b : J → J given by la,bc = abc, ka,bc = acb + bca. Thus, Eq. (1.2) is equivalent to [ ]= − lu,v,lx,y llu,vx,y lx,lv,uy (1.4) for any u, v, x, y. Definition 1.5. A generalized Jordan triple system J is said to be a (−1, −1) Freudenthal Kantor triple system ((−1, −1)-FKTS, for short) if it satisfies + = ld,cka,b ka,blc,d kka,bc,d (1.6) for any a,b,c,d ∈ J . The more general concept of (ε, δ) Freudenthal Kantor triple system was introduced in [23] for ε, δ =±1. A. Elduque et al. / Journal of Algebra 294 (2005) 19–40 21 In case there is a nonzero symmetric bilinear form .|. defined on a (−1, −1)-FKTS such that ka,bc = 2a|bc (1.7) for any a,b,c,the(−1, −1)-FKTSissaidtobebalanced. In this case, Eqs. (1.6) and (1.7) simplify to xxy = xyx =x|xy (1.8) for any x,y. Therefore, a (−1, −1)-balanced Freudenthal Kantor triple system (or (−1, −1)- BFKTS, for short) is a vector spaced over a field F endowed with a trilinear operation xyz and a nonzero symmetric bilinear form .|. such that both (1.2) and (1.8) hold. Some simple (−1, −1)-BFKTS’s were used in [11] to give models of the simple com- plex exceptional Lie superalgebras of classical type: D(2, 1; α), G(3) and F(4). Further- more, this close relationship between (−1, −1)-BFKTS’s and some Lie superalgebras was used in [6] to provide the classification of the simple finite-dimensional (−1, −1)- BFKTS’s over fields of characteristic zero. The aim of this paper is to show that a quadratic noncommutative Jordan algebra in a specific variety can be attached to any (−1, −1)-BFKTS in such a way that the triple product is determined by the (binary) multiplication of the algebra. The classification of the simple finite-dimensional quadratic noncommutative Jordan algebras over fields of characteristic 0 in this variety will be deduced too from the known classification of the simple (−1, −1)-BFKTS’s. The next section will be devoted to introduce the variety V of noncommutative Jordan algebras that will be relevant for our purposes. Then in Section 3, the relationship between some GJTS’s and algebras in V will be studied, while in Section 4 the attention will be restricted to (−1, −1)-BFKTS’s. Section 6 will deal with the classification of the simple finite-dimensional quadratic noncommutative Jordan algebras in the variety V over fields of characteristic 0. 2. A variety of noncommutative Jordan algebras Given any algebra A over a field F (always of characteristic = 2), let Lx and Rx denote the left and right multiplications by x: Lx(y) = xy, Rx(y) = yx; and let (x,y,z) denote the associator of the elements x,y,z: (x,y,z)= (xy)z − x(yz). Recall that the algebra A is a noncommutative Jordan algebra if it is flexible: (x,y,x)= 0 (2.1) for any x,y ∈ A, and satisfies the Jordan identity x,y,x2 = 0 (2.2) 22 A. Elduque et al. / Journal of Algebra 294 (2005) 19–40 for any x,y ∈ A. [ ]= ∈ The Jordan identity is equivalent to the condition Lx,Rx2 0 for any x A, while the flexibility amounts to [Lx,Rx]=0 for any x ∈ A. Also, by flexibility (x,x,y)+ (y,x,x)= 0 ∈ − 2 = − 2 for any x,y A,orLx2 Lx Rx2 Rx . Thus, in the presence of flexibility, [ ]= Lx,Rx2 0 if and only if [ ]= Lx,Lx2 0 (2.3) for any x ∈ A. Given two elements x,y of a flexible algebra A, consider the linear map A → A given by Dx,y = L[x,y] −[Lx,Ly]=(x, y, . ) − (y, x, . ) = (.,x,y)− (.,y,x) =−R[x,y] −[Rx,Ry]. (2.4) = Notice that (2.3) is equivalent to the condition Dx2,x 0. Let V be the variety of those noncommutative Jordan algebras A over a field F satisfying that Dx,y is a derivation of A for any x,y ∈ A. (2.5) This is the variety that will be relevant in what follows. It must be noted that the unital conservative algebras considered in [13] belong to V, because of [13, Proposition 2] and [10, Example 7]. Theorem 2.6. Let A be an algebra in V, then for any x,y,z ∈ A, Dxy,z + Dyz,x + Dzx,y = 0. (2.7) Proof. The fact that Dx,y = L[x,y] −[Lx,Ly]∈Der A (the Lie algebra of derivations of A) for any x,y ∈ A is equivalent to the validity of [L[x,y],Lz]− [Lx,Ly],Lz = L(x,y,z)−(y,x,z) for any x,y,z ∈ A. Permute cyclically x,y,z and add the resulting equations to get [L[x,y],Lz]+[L[y,z],Lx]+[L[z,x],Ly] = L(x,y,z)+(y,z,x)+(z,x,y)−(y,x,z)−(z,y,x)−(x,z,y). (2.8) But in any algebra, A. Elduque et al. / Journal of Algebra 294 (2005) 19–40 23 (x,y,z)+ (y,z,x)+ (z,x,y)− (y,x,z)− (z,y,x)− (x,z,y) = [x,y],z + [y,z],x + [z, x],y , so (2.8) is equivalent to [L[x,y],Lz]+[L[y,z],Lx]+[L[z,x],Ly]=L[[x,y],z]+[[y,z],x]+[[z,x],y], or to D[x,y],z + D[y,z],x + D[z,x],y = 0. (2.9) = ∈ But Dx2,x 0 for any x A, since A is a noncommutative Jordan algebra, so that, by + = ∈ ◦ = + linearization, Dx◦y,x Dx2,y 0 for any x,y A, where x y xy yx, and Dx◦y,z + Dy◦z,x + Dz◦x,y = 0 (2.10) for any x,y,z ∈ A. The result now follows by adding up (2.9) and (2.10). 2 The algebras A endowed with a skew symmetric bilinear map D : A × A → Der A, (x, y) → Dx,y satisfying (2.7) have been named generalized structurable algebras in [10]. The variety of generalized structurable algebras includes the most usual varieties of nonas- sociative algebras. Corollary 2.11. AnyalgebrainV is a generalized structurable algebra. 3. Generalized Jordan triple systems It will be shown in this section the close connection of the algebras in the variety V with some generalized Jordan triple systems.

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