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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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.
(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 specied.
In order to render this paper as self-contained as possible, we recall rst the denition 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 × V → V , is said to be a GJTS of 2nd order if the following conditions are fullled:
Definition 1.2. A Jordan triple system (for short JTS) satises (1.1) and (abc) = (cba),.
Definition 1.3. For ε = ±1 and δ = ±1, a triple product that satises the identities
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, bi ∈ U 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), xk ∈Vk, 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: . U is called simple if ( , , ) is not a zero map and U has no nontrivial ideal.
We denote the triple products by 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
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
where L(a,b) is dened 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.
Definition 1.10. For δ = ±1, atriple system is called a -Lie triple system (for short δ-LTS) if the following identities are fullled:
where . An 1-LTS is a LTS, while a –1-LTS is called an anti-LTS, by .
Corollary 1.12. Let U (ε, δ) be an (ε, δ) )-FKTS. Then the vector space T (ε, δ) = U (ε, δ) U (ε, δ) becomes an LTS (if δ = 1) or an anti-LTS (if δ = –1) with respect to the triple product (1.10):
Remark 1.13. Thus we can obtain the standard embedding Lie algebra (if δ = 1) or Lie superalgebra (if δ = –1), , 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.,
Remark 1.14. is the 5-graded Lie (super)algebra such that . This Lie (super)algebra construction is one of the reasons to study nonassociative algebras and triple systems.
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 dene 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 dened δ-structurable algebras (see ) 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 , the anti-structurable algebras (see ) 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., for x, y ∈ A) over Φ. The identity element of A is denoted by 1.
Remark 1.15. By  we have .
Suppose . Note that (1.11) is valid.
Let Lx, Rx be defined by , define (1.12) and (1.13).
Definition 1.16. is called the triple system obtained from the algebra (A,– ). We call the anti-triple system obtained from the algebra (A,– ).
We will write for short .
Remark 1.17. The upper left index notation is chosen in order not to be mixed with the upper right index notation of  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:
for , and we will call (A,– ) an anti-structurable algebra if identity (1.14) is fullled for .
Definition 1.20. If (A,– ) is anti-structurable, then we call BA an anti-GJTS.
Put . Then, by (1.12),.
Remark 1.21. (i) If u = and x, y ∈ A, (1.14) becomes (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 . Conversely, by [24, Section 3], any Jordan algebra satises (1.15) if hence it is structurable. By (1.15) and , any Jordan algebra is anti-structurable if it satisfies for .
Clearly, the last identity is fullled by an associative algebra.
(iii) If x ∈ A and Tx(1) = 0, then x = 0, by .
Definition 1.22. For and , we say that (A, – ) is skew-alternative if [s, x, y] = –[x, s, y] while (A, – ) is skew-alternative if [h, x, y] = –[x, h, y] for x, y ∈ A.
Remark 1.23. If (A, – ) is skew-alternative, then by , , x, y ∈ A If (A, – ) is skew-alternative, then by (1.11), , , x, y ∈ A.
Proposition 1.24 (see ). If (A, – ) is structurable, then (A, – ) is skew-alternative. If (A, – ) is anti-structurable, then (A, – ) is 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 and so we may dene the structure algebra . This algebra plays an important role in the structure study of structurable algebras (see ) and may play a role in the structure study of anti-structurable algebras (theory to be presented elsewhere).
Definition 1.26. Let (B, U) and be two triple systems. A linear map μ of U into is called a homomorphism if μ satisfies . Moreover, if μ is bijective, then μ is called an isomorphism and (B, U) and 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 dened by , 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 map.
Let Lx, Rx be defined by , define (1.16) and (1.17).
Proposition 1.29. A is a δ-structurable algebra if and only if Aop is a δ-structurable algebra.
Proof. Clearly, is the triple system obtained from the algebra (Aop, – ), and so BA and are isomorphic under the map , by (1.13) and (1.17).
Let denote the vector space of m × n matrices over Φ and for denote by the transposed matrix.
Lemma 1.30. is a (–1, δ)-FKTS, where is dened by (1.18).
Proof. It is straightforward calculation to show that the identities (1.3) and (1.4) hold.
Theorem 1.31. with the involution is a δ-structurable algebra.
Proof. It is a direct consequence of Lemma 1.30.
Example 1.32. is a (–1, δ)-FKTS, where is dened by (1.19).
Indeed, it is straightforward calculation to show that the identities (1.3) and (1.4) hold. Hence with the involution is a δ-structurable algebra.
Remark 1.33. By , the following construction of Lie superalgebras is obtained by the standard embedding method. If with the product (1.18), then the corresponding standard embedding Lie superalgebra is (as dened by [8,9]), hence the standard embedding Lie superalgebra of the anti-structurable algebra . Similarly, if with the product (1.18), then the corresponding standard embedding Lie superalgebra is (as dened by [8,9]), hence the standard embedding Lie superalgebra of the anti-structurable algebra .
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 dened 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 with the product (1.18) and is dened by (2.1)
Then from the previous section this triple system is a simple unitary (–1, –1)-FKTS obtained from anti-structurable algebra . 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 .
Proposition 2.1. Let be anti-structurable algebras and let L(U) = be the standard embedding Lie superalgebra. Then and the corresponding extended Dynkin diagrams with roots deleted are
Proof. From and somewhat long calculations, it follows that defined by (2.1) are simple unitary (–1, –1)-FKTSs. Then the standard embedding Lie superalgebras follows from . Moreover, since is an anti-LTS and , it is a straightforward calculation to check that is obtained from the extended Dynkin diagram of L(U) by deleting the root , while L0 is isomorphic to the corresponding Dynkin diagram ( deleted) .
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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