ON LIE ALGEBRAS AND TRIPLE SYSTEMS N. Kamiya Department of Mathematics, University of Aizu, Aizuwakamatsu, 965-8580, Japan e-mail: [email protected]

Abstract In this paper, we consider several different constructions of simple B3 -type Lie algebras from several triple systems and the correspondence with extended Dynkin diagrams associated with such triple systems.

Introduction In the future, it is expected that triple systems will be useful for the characterization of noncommutative structures in mathematics and physics as well as that of (classical) Yang-Baxter equations [9,17,18,20,21]. Our aim is to use triple systems to investigate a characterization of differential geometry and mathematical physics from the viewpoint of nonassociative algebras that contain a class of Lie algebras or Jordan algebras [7,8,14,16,17,20]. Thus, in particular, for B3 -type Lie algebras, we will provide some examples of triple systems and their correspondence with extended Dynkin diagrams in this article. A (2ν + 1) graded Lie algebra is a Lie algebra of the form g = ⊕νk=−ν gk such that [gk , gl ] ⊂ gk+l . It is well-known that 3-graded Lie algebras are essentially in bijection with certain theoretic objects called Jordan pairs. Kantor remarked that more general graded Lie algebras correspond to generalized Jordan triple systems. In particular, the graded Lie algebra g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 has the structure of a triple product on the subspace g−1 , and is known as a generalized Jordan triple system (GJTS) of second order or a (-1,1)-Freudenthal-Kantor triple system (F-K.t.s.) [7,8,12]. Also g−1 ⊕ g1 has the structure of a Lie triple system (in particular, a system over a real number is known to correspond with a symmetric Riemannian space). We will discuss the corresponding geometrical object by means of these triple systems. The notation and terminology used for the geometry can be found in [4,5,19]. We will often use the symbols g and L to denote a Lie algebra or Lie superalgebra as is conventionally done [2,3,6,23]. Speaking from the viewpoint of an algebraic study, our purpose is to propose a unified structural theory for triple systems in nonassociative algebras. In previous works [11,12,13], we have studied the Peirce decomposition of the GJTS U of second order by employing a tripotent element e of U ( for a tripotent element, {eee} = e). The Peirce decomposition of U is described as follows: 201

U = U00 ⊕U 1 1 ⊕U11 ⊕U 3 3 ⊕U− 1 0 ⊕U01 ⊕U 1 2 ⊕U13 , 22

22

2

2

where L(a) = {eea} = λa and R(a) = {aee} = µa if a ∈ Uλµ . This point of view has been the basis of our study on triple systems. We consider triple systems which are finite dimensional over a field Φ of characteristic 6= 2 or 3, unless specified otherwise. This note is an announcement of new results, and the details will be published elsewhere.

1

Preliminaries

To make this paper as self-contained as possible, we first recall the definition of a generalized Jordan triple system of second order (hereafter, referred to as the GJTS of 2nd order), and the construction of Lie algebras associated with GJTS of 2nd order. A vector space V over a field Φ, endowed with a trilinear operation V ×V ×V → V , (x, y, z) 7−→ {xyz}, is said to be a GJTS of second order if the following two conditions are satisfied: (J1) {ab{xyz}} = {{abx}yz} − {x{bay}z} + {xy{abz}} (GJT S) (K1) K(K(a, b)x, y) − L(y, x)K(a, b) − K(a, b)L(x, y) = 0 (2nd order), where L(a, b)c = {abc} and K(a, b)c = {acb} − {bca}. Remark 1.1. If K(a, b) ≡ 0 (identically zero), then this triple system is a Jordan triple system (JTS), i.e., it satisfies the relations {acb} = {cba} and GJT S. We can also generalize the concept of the GJTS of 2nd order as follows (see for example [7,8,10,14] and the references therein). For ε = ±1 and δ = ±1, if the triple product satisfies (ab(xyz)) = ((abx)yz) + ε(x(bay)z) + (xy(abz)), K(K(a, b)c, d) − L(d, c)K(a, b) + εK(a, b)L(c, d) = 0, where L(x, y)z = (xyz) and K(a, b)c = (acb) − δ(bca), then it is said to be a (ε, δ)-FreudenthalKantor triple system, shortly a (ε, δ)-F-K.t.s. If, moreover, dimΦ {K(a, b)}span = 1, then we say that the(ε, δ)-F-K.t.s is said to be balanced. Remark 1.2. S(x, y) := L(x, y) + εL(y, x), and A(x, y) := L(x, y) − εL(y, x), are respectively a derivation and an anti-derivation of U(ε, δ). We generally denote the triple products by {xyz}, (xyz), [xyz], and < xyz >. Bilinear forms are denoted by < x|y >, (x, y), and B(x, y). Remark 1.3. Note that the concept of a GJTS of 2nd order coincides with that of (−1, 1)-FK.t.s. Thus we can construct simple Lie algebras or superalgebras by means of the standard embedding method (see for example [2,3,7-11,13,14,15,21]. Theorem 1.4. ([8,15]). Let U(ε, δ) be an (ε, δ)-F-K.t.s. If J is an endomorphism of U(ε, δ) such that J < xyz >=< JxJyJz > and J 2 = −εδId, then (U(ε, δ), [xyz]) is a Lie triple system (the case of δ = 1) or an anti-Lie triple system (the case of δ = −1) with respect to the product [xyz] :=< xJyz > −δ < yJxz > +δ < xJzy > − < yJzx > . 202

Corollary 1.5. ([7]). Let U(ε, δ) be an (ε, δ)-F-K.t.s. Then the vector space T (ε, δ) = U(ε, δ)⊕ U(ε, δ) becomes a Lie triple system (the case of δ = 1) or an anti-Lie triple system (the case of δ = -1) with respect to the triple product defined by        L(a, d) − δL(c, b) δK(a, c) e a c e [ ]= . −εK(b, d) ε(L(d, a) − δL(b, c)) f b d f Thus we can obtain the standard embedding Lie algebra (the case of δ = 1) or Lie superalgebra (the case of δ = −1), L(ε, δ) = D(T (ε, δ), T (ε, δ)) ⊕ T (ε, δ), associated with T (ε, δ), where D(T (ε, δ), T (ε, δ)) is the set of inner derivations of T (ε, δ). That is, these vector spaces D(T (ε, δ), T (ε, δ)) and T (ε, δ) imply   L(a, b) δK(c, d) D(T (ε, δ), T (ε, δ)) := and −εK(e, f ) εL(b, a) span   x T (ε, δ) := { |x, y ∈ U(ε, δ)}span . y In fact, we have   L(a, b) 0 L0 = { } = {L(a, b)}span , 0 εL(b, a) span   0 δK(c, d) L−2 = { }span = {K(c, d)}span and L0 = Der U ⊕ Anti-Der U. 0 0 Remark 1.6. For the standard embedding algebras obtained from these triple systems, note that L(ε, δ) := L−2 ⊕ L−1 ⊕ L0 ⊕ L−1 ⊕ L−2 (or g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ) is a 5-graded Lie algebra or Lie superalgebra, such that L−1 = g−1 = U(ε, δ) and Der T (U) := D(T (ε, δ), T (ε, δ)) = L−2 ⊕ L0 ⊕ L−2 with [Li , L j ] ⊆ Li+ j . By straightforward calculations, for the correspondence of the (1,1) balanced F.K.t.s with the (-1,1) balanced F.K.t.s, we obtain the following. Theorem 1.7. Let (U, < xyz >) be a (1, 1) F-K.t.s. If there is an endomorphism J of U such that J < xyz >=< JxJyJz > and J 2 = −Id, then (U, {xyz}) is a GJTS of 2nd order (that is, (-1,1)-F-K.t.s.) with respect to the new product defined by {xyz} :=< xJyz > . We now give an explicit example of a JTS and a Lie triple system. Example 1.8. Let U be a vector space with a symmetric bilinear form < , >. Then the triple system (U, [xyz]) is a Lie triple system with respect to the product [xyz] =< y, z > x− < z, x > y. That is, this triple system is induced from the JTS 1 {xyz} = (< x, y > z+ < y, z > x− < z, x > y), 2 by means of [xyz] = {xyz} − {yxz}. 203

2

Construction of B3-type Lie algebras from several triple systems

In this section, we will discuss the construction of simple B3 -type Lie algebras associated with several triple systems (the details will be described in a future paper). a) the case of a JTS, b) the case of a balanced GJTS, c) the case of a GJTS of 2nd order, d) the case of a derivation induced from a JTS. To consider these cases, we will start with an extended Dynkin diagram for the B3 -type Lie algebra. 1 2 2 ◦ · · · ◦ => ◦ | ◦ −ρ where −ρ =α1 + 2α2 + 2α3 . For the root system, it is well known that {α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 , α2 + 2α3 , α1 + α2 + 2α3 , α1 + 2α1 + 2α1 }.

2.1

The case of a JTS

First we study the case of g−1 = U = Mat(1, 5; Φ). (Hereafter, we assume Φ=C.) In this case, g−1 is a JTS respect to the product {xyz} = xt yz + yt zx − zt xy, where t x denotes the transpose matrix of x. By straightforward calculations, the standard embedding Lie algebra L(U) = g can be shown to be 3-graded B3 -type Lie algebra with g−1 ⊕ g0 ⊕ g1 . Thus, we have   Id 0 g0 = Der U ⊕ Anti-Der U = B2 ⊕ ΦH, where H := 0 −Id Der(g−1 ⊕ g1 ) ∼ = {◦ · · · ◦ =⇒ ◦} = B3 , ( omitted). 1 ···

2 2 ◦ => ◦ | ◦ −ρ

Furtheremore, we obtain DerU = {L(x, y) − L(y, x)}span = B2 , Anti-Der U = {L(x, y) + L(y, x)}span = ΦH,   L(x, y) 0 g0 = { } = {S(x, y) + A(x, y)}span , 0 −L(y, x) span 204

where S(x, y) = L(x, y) − L(x, y), A(x, y) = L(x, y) + L(y, x). Here, g−1 corresponds to the root system {α1 , α1 + α2 , α1 + α2 + α3 , α1 + α2 + 2α3 , α1 + 2α2 + 2α3 }

2.2

The case of a balanced GJTS

Secondly, we study the case of g−1 = U = Mat(2, 3; Φ). In this case, g−1 is a balanced GJTS of 2nd order w.r.t. the product  {xyz} := zt yx + xt yz − zJ3 t xyJ3 , where J3 = 

 1 1 .

1 Straightforward calculations show that L(U) = g is a 5-graded B3 -type Lie algebra with g−2 ⊕ · · · ⊕ g2 and dim g−2 = 1. Thus, we have   Id 0 g0 = Der U ⊕ Anti-Der U = A1 ⊕ A1 ⊕ ΦH where H := 0 −Id Der (g1 ⊕ g1 ) = g−2 ⊕ g0 ⊕ g2 = A1 ⊕ A1 ⊕ A1 ( omitted) ∼ = Der T (U) 1 ◦ ···

2 2 => ◦ | ◦ −ρ

Furthermore we obtain that g−2 = {K(x, y)}span = ΦId · · · , which is one-dimensional, i.e., balanced. This g−1 corresponds to the root system {α2 , α1 + α2 , α1 + α2 + α3 , α2 + α3 , α2 + 2α3 , α1 + α2 + 2α3 , } g−2 corresponds to the highest root {α1 + 2α2 + 2α3 }, and g/(g−2 ⊕ g0 ⊕ g2 ) ∼ = T (U) (= g−1 ⊕ g1 ) is the tangent space of a quaternion symmetric space of dimension 12, since T (U) is a Lie triple system associated with g−1 .

2.3

The case of a GJTS of second order

Third we study the case of g−1 = U = Mat(1, 3; Φ). In this case, g−1 is a GJTS of second order with respect to the product {xyz} = xt yz + zt yx − yt xz. Straightforward computations show that L(U) is a 5-graded B3 -type Lie algebra with g−2 ⊕ · · · ⊕ g2 and dim g−2 = 3,   Id 0 g0 = Der U ⊕ Anti-Der U = A2 ⊕ ΦH where H := 0 −Id 205

Der (g−1 ⊕ g1 ) = g−2 ⊕ g0 ⊕ g2 = A3 ( omitted) ∼ = Der T (U). 1 ◦ ···

2 2 ◦ => | ◦ −ρ

Furthermore, we obtain g−2 = {K(x, y)}span = Alt(3, 3; Φ). That is, the triple system g−1 (resp. g−2 ) corresponds to th the root system {α3 , α2 + α3 , α1 + α2 + α3 } (resp. {α2 + 2α3 , α1 + α2 + 2α3 , α1 + 2α2 + 2α3 }), implying that ◦| · ·{z · · · · ◦} · · · =⇒ ( omitted) and g0 = A2 ⊕ ΦH.

Remark 2.1. Following [8], for the case of a GJT of 2nd order, note that g−2 (∼ = k) has the structure of the JTS associated with a GJTS of second order.

2.4

The case of a derivation induced from a JTS

Finally, we study the case of g−1 = U = Mat(1, 7; Φ). In this case g−1 is a JTS with respect to the product {xyz} = xt yz + yt zx − zt xy. For this case, we obtain Der U = {L(x, y) − L(y, x)}span = Alt(7, 7; Φ) ∼ = B3 , ∼ Anti-Der U = ΦH (which is one dimensional). The standard embedding Lie algebra is a 3-graded B4 -type Lie algebra with g−1 ⊕ g0 ⊕ g1 . Furthermore, we have · · · ◦| · · · ◦{z=⇒ ◦} ( omitted) g0 = B3 ⊕ ΦH. This case is obtained from Der U such that U = Mat(1, 7; Φ) with the JTS structure. Remark 2.2. In the above constructions, note that there exist four different constructions for the B3 -type Lie algebras. It seems that these results may be applicable to mathematical physics, for example, quark theory and gravity theory.

3

Concluding Remarks

3.1. The inner structures of triple systems are closely related to the characterization of root systems of Lie algebras, in particular, we have the following: (i) There exists a correspondence between simple balanced (-1,1)-Freudenthal-Kantor triple systems and quaternionic Riemannian symmetric spaces [1,15]. That is, there exists a correspondence between simple balanced GJTS of 2nd order and quaternionic Riemannian symmetric spaces. 206

(ii) There exists a relationship between Lie triple systems and totally geodesic manifolds [5,19]. (iii) There exists a relationship between symmetric domains and positive definite Hermitian Jordan triple systems [22]. Thus, triple systems appear to be useful tool and concept for characterizing geometrical phenomena. 3.2. For the theory of Peirce decompositions, we refer the reader to [11,12,13]. We will provide examples of special cases of four B3 -type Lie algebras construction in this note in a forthcoming paper. It appears that this concept is to study the inner structure of the triple system U.

Acknowledgements I would like to thank the organizers of the conference “Noncommutative Structures in Mathematics and Physics”, Brussels, July 22-26, 2008, and, in particular, Prof. Stefaan Caenepeel for his hospitality.

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