Medical, Pharma, Engineering, Science, Technology and Business

^{a}Center for Mathematical Sciences, University of Aizu, Aizuwakamatsu, 965-8580 Fukushima, Japan. **E-mail:** [email protected]

^{b}Kungliga Tekniska hogskolan, Valhallavagen 79, 100 44 Stockholm, Sweden. **E-mail:** [email protected]

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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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:

(1.1)

(1.2)

where

**Definition 1.2.** A Jordan triple system (for short JTS) satises (1.1) and (*abc*) = (*cba*),.

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 satises the identities

(1.3)

(1.4)

where

(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 *a _{i}*,

Let *U* be an (*ε*, *δ*)-FKTS and let *V _{k}*,

**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

(1.6a)

(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

where *L*(*a,b*) is dened by (1.5).

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

(1.7)

(1.8)

**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 [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 that is an LTS (if δ = 1) or an anti-LTS (if δ = –1) with respect to the product (1.9):*

(1.9)

**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):*

(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.

*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 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 [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 , 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., for *x, y* ∈ *A*) over Φ. The identity element of *A* is denoted by 1.

**Remark 1.15.** By [1] we have .

Suppose . Note that (1.11) is valid.

(1.11)

Let *L _{x}*,

(1.12)

(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 [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:

(1.14)

for , and we will call (*A*,^{–} ) an anti-structurable algebra if identity (1.14) is fullled for .

**Remark 1.19.** If (*A*,^{–} ) is structurable, then, in the terminology of [21], the triple system *B _{A}* is called a GJTS and by [7],

**Definition 1.20.** If (*A*,^{–} ) is anti-structurable, then we call *B _{A}* an anti-GJTS.

Put . Then, by (1.12),.

**Remark 1.21.** (i) If *u* = and *x*, *y* ∈ *A*, (1.14) becomes (1.15).

(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 satises (1.15) if hence it is structurable. By (1.15) and [18], 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 *T _{x}*(1) = 0, then

**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 [1], , *x*, *y* ∈ *A* If (*A*, ^{–} ) is skew-alternative, then by (1.11), , , *x*, *y* ∈ *A*.

**Proposition 1.24** (see [18]). *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 [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 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 (A^{op},^{ –} ) 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 (A^{op},^{ –} ) are isomorphic under the map.

Let *L _{x}, R_{x}* be defined by , define (1.16) and (1.17).

(1.16)

(1.17)

**Proposition 1.29.** *A* is a *δ*-structurable algebra if and only if *A ^{op}* is a

**Proof.** Clearly, is the triple system obtained from the algebra (*A ^{op}*,

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).*

(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).

(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 [17], 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)

(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 [8].

**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 [17]. 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 *L _{0}* 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.

- Allison BN (1978) A class of nonassociative algebras with involution containing the class of Jordan algebras. Math Ann 237: 133-156.
- Allison BN (1979) Models of isotropic simple Lie algebras. Comm Algebra 7: 1835-1875.
- Asano H, Yamaguti K (1980) A construction of Lie algebras by generalized Jordan triple systemsof second order. Nederl Akad Wetensch Indag Math 42: 249-253.
- Asano H (1991) Classification of non-compact real simple generalized Jordan triple systems of the second kind. Hiroshima Math J 21: 463-489.
- Elduque A, KamiyaN, OkuboS (2003) Simple (1;1) balanced Freudenthal Kantor triplesystems. Glasg Math J 11: 353-372.
- Elduque A, Kamiya N,OkuboS (2005) (1;1) balanced Freudenthal Kantor triple systemsand non-commutative Jordan algebras. J Algebra 294: 19-40.
- Faulkner JR (1994) Structurable triples, Lie triples, and symmetric spaces. Forum Math 6: 637-650.
- Frappat L, SciarrinoA, SorbaP (2000) Dictionary on Lie Algebras and Superalgebras. Academic Press San Diego, California.
- Kac VG (1977) Lie superalgebras. Adv Math 26: 8-96.
- KamiyaN (1987) A structure theory of Freudenthal-Kantor triple systems. J Algebra 110: 108-123.
- Kamiya N (1988) A construction of anti-Lie triple systems from a class of triple systems. Mem Fac Sci Shimane Univ 22: 51-62.
- KamiyaN (1989) A structure theory of Freudenthal-Kantor triple systems. II. Comment. Math Univ St Paul 38: 41-60.
- Kamiya N (1991) The construction of all simple Lie algebras over C from balanced Freudenthal-Kantortriple systems. In \Contributions to General Algebra, 7". D. Dorninger, G. Eigenthaler, H. K.Kaiser, and W. B. Muller, Eds. Holder-Pichler-Tempsky, Vienna pp: 205-213.
- Kamiya N (2005) Examples of Peirce decomposition of generalized Jordan triple system of secondorder-Balanced cases. In \Noncom mutative Geometry and Representation Theory in MathematicalPhysics". J. Fuchs, J. Michelson, G. Rozenblioum, A. Stolin, and A. Westerberg, Eds.Contemp. Math. 391, American Mathematical Society, Providence, RI pp: 157-165.
- KamiyaN, OkuboS (2000) On -Lie supertriple systems associated with ( ; )-Freudenthal-Kantorsupertriple systems. Proc Edinburgh Math Soc43: 243-260.
- Kamiya N, OkuboS (2003) Construction of Lie superalgebras D(2; 1; ), G(3) and F(4) fromsome triple systems. Proc. Edinburgh Math Soc46: 87-98.
- Kamiya N, Okubo S (2004) A construction of simple Lie superalgebras of certain types from triple systems. Bull Austral Math Soc 69: 113-123.
- Kamiya N, MondocD (2008) A new class of nonassociative algebras with involution. Proc JapanAcad Ser A 84: 68-72.
- Kaneyuki S, AsanoH (1988) Graded Lie algebras and generalized Jordan triple systems. Nagoya Math J 112: 81-115.
- KantorIL (1970) Graded Lie algebras. Trudy Sem Vect Tens Anal 15: 227-266.
- Kantor IL (1972) Some generalizations of Jordan algebras. Trudy Sem Vect Tens Anal 16: 407-499.
- Kantor IL (1973) Models of exceptional Lie algebras. Soviet Math Dokl 14: 254-258.
- Kantor IL, KamiyaN (2003) A Peirce decomposition for generalized Jordan triple systems ofsecond order. Comm Algebra 31: 5875-5913.
- Koecher M (1967) Embedding of Jordan algebras into Lie algebras I. Amer J Math 89: 787-816.
- Koecher M (1968) Embedding of Jordan algebras into Lie algebras II. Amer J Math 90: 476-510.
- Mondoc D (2006) Models of compact simple Kantor triple systems de ned on a class of structurablealgebras of skew-dimension one. Comm Algebra 34: 3801-3815.
- Mondoc D (2007) On compact reali cations of exceptional simple Kantor triple systems. J Gen Lie Theory Appl 1: 29-40.
- Mondoc D (2007) Compact reali cations of exceptional simple Kantor triple systems de ned on tensorproducts of composition algebras. J Algebra 307: 917-929.
- Mondoc D (2007) Compact exceptional simple Kantor triple systems de ned on tensor products ofcomposition algebras. Comm Algebra 35: 3699-3712.
- Okubo S (2005) Symmetric triality relations and structurable algebras. Linear Algebra Appl 396: 189-222.
- Tits J (1962) Une classe d'alg ebres de Lie en relation avec les alg ebres de Jordan. Nederl Acad Wetensch Proc Ser A 65: 530-535.
- Yamaguti K, OnoA (1984) On representations of Freudenthal-Kantor triple systems U( ; ), Bull. Fac. School Ed. Hiroshima Univ 7: 43-51

Select your language of interest to view the total content in your interested language

- Adomian Decomposition Method
- Algebra
- Algebraic Geometry
- Analytical Geometry
- Applied Mathematics
- Axioms
- Balance Law
- Behaviometrics
- Big Data Analytics
- Binary and Non-normal Continuous Data
- Binomial Regression
- Biometrics
- Biostatistics methods
- Clinical Trail
- Combinatorics
- Complex Analysis
- Computational Model
- Convection Diffusion Equations
- Cross-Covariance and Cross-Correlation
- Deformations Theory
- Differential Equations
- Differential Transform Method
- Fourier Analysis
- Fuzzy Boundary Value
- Fuzzy Environments
- Fuzzy Quasi-Metric Space
- Genetic Linkage
- Geometry
- Hamilton Mechanics
- Harmonic Analysis
- Homological Algebra
- Homotopical Algebra
- Hypothesis Testing
- Integrated Analysis
- Integration
- Large-scale Survey Data
- Latin Squares
- Lie Algebra
- Lie Superalgebra
- Lie Theory
- Lie Triple Systems
- Loop Algebra
- Matrix
- Microarray Studies
- Mixed Initial-boundary Value
- Molecular Modelling
- Multivariate-Normal Model
- Noether's theorem
- Non rigid Image Registration
- Nonlinear Differential Equations
- Number Theory
- Numerical Solutions
- Operad Theory
- Physical Mathematics
- Quantum Group
- Quantum Mechanics
- Quantum electrodynamics
- Quasi-Group
- Quasilinear Hyperbolic Systems
- Regressions
- Relativity
- Representation theory
- Riemannian Geometry
- Robust Method
- Semi Analytical-Solution
- Sensitivity Analysis
- Smooth Complexities
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Super Algebras
- Symmetric Spaces
- Theoretical Physics
- Theory of Mathematical Modeling
- Three Dimensional Steady State
- Topologies
- Topology
- mirror symmetry
- vector bundle

- 7th International Conference on Biostatistics and Bioinformatics

September 26-27, 2018 Chicago, USA - Conference on Biostatistics and Informatics

December 05-06-2018 Dubai, UAE - Mathematics Congress - From Applied to Derivatives

December 5-6, 2018 Dubai, UAE

- Total views:
**11547** - [From(publication date):

September-2008 - Jun 23, 2018] - Breakdown by view type
- HTML page views :
**7773** - PDF downloads :
**3774**

Peer Reviewed Journals

International Conferences 2018-19