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On anti-structurable algebras and extended Dynkin diagrams | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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On anti-structurable algebras and extended Dynkin diagrams

Noriaki KAMIYA a and Daniel MONDOC b

aCenter for Mathematical Sciences, University of Aizu, Aizuwakamatsu, 965-8580 Fukushima, Japan. E-mail: [email protected]

bKungliga Tekniska hogskolan, Valhallavagen 79, 100 44 Stockholm, Sweden. E-mail: [email protected]

Received Date: September 01, 2008; Revised Date: December 09, 2008

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Abstract

We construct Lie superalgebras osp(2n + 1 j 4n + 2) and osp(2n j 4n) starting with certain classes of anti-structurable algebras via the standard embedding Lie superalgebra construction corresponding to (; )-Freudenthal Kantor triple systems.

Introduction

(ε, δ)-Freudenthal Kantor triple systems, δ-Lie triple systems, and Lie (super)algebras

We are concerned in this paper with triple systems which have nite dimension over a eld Φ of characteristic ≠ 2 or 3, unless otherwise speci ed.

In order to render this paper as self-contained as possible, we recall rst the de nition of a generalized Jordan triple system of second order (for short GJTS of 2nd order).

Definition 1.1. A vector space V over a eld Φ endowed with a trilinear operation V × V × VV , Equation is said to be a GJTS of 2nd order if the following conditions are ful lled:

Equation    (1.1)

Equation    (1.2)

where

Equation

Definition 1.2. A Jordan triple system (for short JTS) satis es (1.1) and (abc) = (cba),Equation.

We can generalize the concept of GJTS of the 2nd order as follows (see [10,11,13,15,32]).

Definition 1.3. For ε = ±1 and δ = ±1, a triple product that satis es the identities

Equation    (1.3)

Equation    (1.4)

where

Equation    (1.5)

is called an (ε, δ)-Freudenthal-Kantor triple system (for short (ε, δ)-FKTS).

Remark 1.4. Note that K (b, a) = –δK (a, b).

Definition 1.5. An (ε, δ)-FKTS U is called unitary if the identity map I d is contained in κ := K(U,U), i.e., if there exist ai, biU such that ΣiK(ai, bi) = I d.

Let U be an (ε, δ)-FKTS and let Vk, k = 1, 2, 3, be subspaces of U. We denote by (V1, V2, V3) the subspace of U spanned by elements (x1, x2, x3), xkVk, k = 1, 2, 3.

Definition 1.6. A subspace V of U is called an ideal of an (ε, δ)-FKTS U if the following relations hold: Equation. U is called simple if ( , , ) is not a zero map and U has no nontrivial ideal.

We denote the triple products by Equation upon their suitability.

Remark 1.7. We note that the concept of GJTS of 2nd order coincides with that of (–1; 1)- FKTS. Thus we can construct the simple Lie algebras by means of the standard embedding method (see [5,10-13,15-17,22,32]).

Remark 1.8. We note that the two pairs of identities (1.3-1.4) and (1.6) are equivalent

Equation    (1.6a)

Equation    (1.6b)

where ε = ±1, δ = ±1 and L(a,b),K(a,b) are defined by (1.5).

Indeed, from (1.3) and (1.4) follows (1.6b). Conversely, from (1.6a) and (1.6b) it follows that (1.4) holds.

For an (ε, δ)-FKTS U , we denote

Equation

where L(a,b) is de ned by (1.5).

Remark 1.9. We note that S(a,b) = εS(b,a).

Then S(a,b) (resp., A(a,b)) is a derivation (resp., anti-derivation) of U. Indeed, we note that the identities (1.7) and (1.8) are valid.

Equation     (1.7)

Equation     (1.8)

Definition 1.10. For δ = ±1, atriple system Equation is called a -Lie triple system (for short δ-LTS) if the following identities are ful lled:

Equation

where Equation. An 1-LTS is a LTS, while a –1-LTS is called an anti-LTS, by [11].

Proposition 1.11 (see [11,15]). Let U (ε, δ) be an (ε, δ) )-FKTS. If J is an endomorphism of U (ε, δ) such thatEquation is an LTS (if δ = 1) or an anti-LTS (if δ = –1) with respect to the product (1.9):

Equation     (1.9)

Corollary 1.12. Let U (ε, δ) be an (ε, δ) )-FKTS. Then the vector space T (ε, δ) = U (ε, δ) Equation U (ε, δ) becomes an LTS (if δ = 1) or an anti-LTS (if δ = –1) with respect to the triple product (1.10):

Equation     (1.10)

Remark 1.13. Thus we can obtain the standard embedding Lie algebra (if δ = 1) or Lie superalgebra (if δ = –1), Equation, associated to T (ε, δ), where D(T (ε, δ),T (ε, δ)) is the set of inner derivations of T (ε, δ),i.e.,

Equation

Remark 1.14. Equation is the 5-graded Lie (super)algebra such that Equation. This Lie (super)algebra construction is one of the reasons to study nonassociative algebras and triple systems.

δ-structurable algebras

The existence of the class of nonassociative algebras called structurable algebras is an important generalization of Jordan algebras giving a construction of Lie algebras. Hence from our concept, by means of triple products, we de ne a generalization of such class to construct Lie superalgebras as well as Lie algebras. Our start point briey described in a historical setting is the construction of Lie (super)algebras starting from a class of nonassociative algebras. Hence within the general framework of (ε, δ)-FKTS (ε, δ = ±1) and the standard embedding Lie (super)algebra construction studied in [5,6,10-12,17] (see also references therein) we de ned δ-structurable algebras (see [18]) as a class of nonassociative algebras with involution which coincides with the class of structurable algebras for δ = 1 as introduced and studied in [1,2]. Structurable algebras are a class of nonassociative algebras with involution that include Jordan algebras (with trivial involution), associative algebras with involution, and alternative algebras with involution. They are related to GJTSs of 2nd order, or (–1, 1)-FKTSs, as introduced and studied in [20,21] and further studied in [3,4,19,26-30] (see also references therein). Their importance lies with constructions of 5-graded Lie algebras Equation, the anti-structurable algebras (see [18]) are a class of nonassociative algebras that may similarly shed light on the notion of (–1;–1)-FKTSs hence, by [5,6], on the construction of Lie superalgebras and Jordan algebras as it will be shown.

Throughout the paper, it is assumed that (A, ) is a nite-dimensional nonassociative unital algebra with involution (involutive anti-automorphism, i.e., Equation for x, yA) over Φ. The identity element of A is denoted by 1.

Remark 1.15. By [1] we have Equation.

Suppose Equation. Note that (1.11) is valid.

Equation     (1.11)

Let Lx, Rx be defined by Equation, define (1.12) and (1.13).

Equation     (1.12)

Equation     (1.13)

Definition 1.16. Equation is called the triple system obtained from the algebra (A, ). We call Equation the anti-triple system obtained from the algebra (A, ).

We will write for short Equation.

Remark 1.17. The upper left index notation is chosen in order not to be mixed with the upper right index notation of [1] which has a different meaning.

Definition 1.18. A unital nonassociative algebra with involution (A, ) is called a structurable algebra if the following identity is fulfilled:

Equation    (1.14)

for Equation, and we will call (A, ) an anti-structurable algebra if identity (1.14) is ful lled for Equation.

Remark 1.19. If (A, ) is structurable, then, in the terminology of [21], the triple system BA is called a GJTS and by [7], BA is a GJTS of 2nd order, i.e., satis es the identities (1.3) and (1.4).

Definition 1.20. If (A, ) is anti-structurable, then we call BA an anti-GJTS.

Put Equation. Then, by (1.12),EquationEquation.

Remark 1.21. (i) If u = Equation and x, yA, (1.14) becomes (1.15).

Equation     (1.15)

(ii) Suppose is the identity map and hence A is commutative. If (A, ) is δ-structurable, then A is a Jordan algebra, by [18]. Conversely, by [24, Section 3], any Jordan algebra satis es (1.15) if Equation hence it is structurable. By (1.15) and [18], any Jordan algebra is anti-structurable if it satisfies Equation for Equation.

Clearly, the last identity is ful lled by an associative algebra.

(iii) If xA and Tx(1) = 0, then x = 0, by [18].

Definition 1.22. For Equation and Equation , we say that (A, ) is Equation skew-alternative if [s, x, y] = –[x, s, y] while (A, ) isEquation skew-alternative if [h, x, y] = –[x, h, y] for x, yA.

Remark 1.23. If (A, ) is Equation skew-alternative, then by [1], EquationEquation, x, yA If (A, ) is Equation skew-alternative, then by (1.11), Equation, Equation, x, yA.

Proposition 1.24 (see [18]). If (A, ) is structurable, then (A, ) is Equation skew-alternative. If (A, ) is anti-structurable, then (A, ) is Equation skew-alternative.

Remark 1.25. Let (A, ) be a δ-structurable algebra and let Der(A, ) be the set of derivations of A that commute with . By Remark (iii) above Equation and so we may de ne the structure algebra Equation. This algebra plays an important role in the structure study of structurable algebras (see [1]) and may play a role in the structure study of anti-structurable algebras (theory to be presented elsewhere).

Examples

For examples of structurable algebras, we refer to [1,2].

Definition 1.26. Let (B, U) and Equation be two triple systems. A linear map μ of U into Equation is called a homomorphism if μ satisfies Equation. Moreover, if μ is bijective, then μ is called an isomorphism and (B, U) and Equation are said to be isomorphic.

Definition 1.27. Let (A, ) be a unital nonassociative algebra over Φ with involution and let (Aop, ) denote the opposite algebra, i.e., the algebra with multiplication de ned by Equation, where in the right-hand side of the equality the multiplication is done in A.

Remark 1.28. The algebras (A, ) and (Aop, ) are isomorphic under the mapEquation.

Let Lx, Rx be defined by Equation, define (1.16) and (1.17).

Equation     (1.16)

Equation    (1.17)

Proposition 1.29. A is a δ-structurable algebra if and only if Aop is a δ-structurable algebra.

Proof. Clearly, Equation is the triple system obtained from the algebra (Aop, ), and so BA and Equation are isomorphic under the map Equation, by (1.13) and (1.17).

Let Equation denote the vector space of m × n matrices over Φ and for Equation denote by Equation the transposed matrix.

Lemma 1.30. Equation is a (–1, δ)-FKTS, where Equation is de ned by (1.18).

Equation     (1.18)

Proof. It is straightforward calculation to show that the identities (1.3) and (1.4) hold.

Theorem 1.31.Equation with the involutionEquation is a δ-structurable algebra.

Proof. It is a direct consequence of Lemma 1.30.

Example 1.32.Equation is a (–1, δ)-FKTS, where Equation is de ned by (1.19).

Equation      (1.19)

Indeed, it is straightforward calculation to show that the identities (1.3) and (1.4) hold. Hence Equation with the involution Equation is a δ-structurable algebra.

Remark 1.33. By [17], the following construction of Lie superalgebras is obtained by the standard embedding method. If Equation with the product (1.18), then the corresponding standard embedding Lie superalgebra is Equation (as de ned by [8,9]), hence the standard embedding Lie superalgebra of the anti-structurable algebra Equation. Similarly, if Equation with the product (1.18), then the corresponding standard embedding Lie superalgebra is Equation (as de ned by [8,9]), hence the standard embedding Lie superalgebra of the anti-structurable algebra Equation.

The construction of these Lie superalgebras and the correspondence with extended Dynkin diagrams is the subject of the next section. The study of the structure theory of antistructurable algebras, the Peirce decomposition (as de ned by [14,23]), will be considered as future work. Moreover, let U be an anti-structurable algebra and associative algebra, then U is a (–1, –1)-FKTS. The details will be described in a future paper.

Anti-structurable algebras and extended Dynkin diagrams

Let Equation with the product (1.18) and Equation is de ned by (2.1)

Equation     (2.1)

Then from the previous section this triple system is a simple unitary (–1, –1)-FKTS obtained from anti-structurable algebra Equation. Hence by the methods of the standard embedding associated to U we can obtain the standard embedding Lie superalgebra as follows from the following proposition: the Lie (super)algebras notations and extended Dynkin diagrams are those of [8].

Proposition 2.1. Let Equation be anti-structurable algebras and let L(U) =Equation be the standard embedding Lie superalgebra. Then Equation and the corresponding extended Dynkin diagrams with Equation roots deleted are

Equation

Proof. From Equation and somewhat long calculations, it follows that Equation defined by (2.1) are simple unitary (–1, –1)-FKTSs. Then the standard embedding Lie superalgebras follows from [17]. Moreover, since Equation is an anti-LTS and Equation, it is a straightforward calculation to check that Equation is obtained from the extended Dynkin diagram of L(U) by deleting the root Equation , while L0 is isomorphic to the corresponding Dynkin diagram (Equation deleted) Equation.

Remark 2.2. These results mean that the correspondence between anti-structurable algebras and extended Dynkin diagrams is a useful concept for the structure theory of triple systems.

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