Theory an ie d L A p d p e l z i i c l a a Davvaza et al., J Generalized Lie Theory Appl 2015, 9:2 t Journal of Generalized Lie r i o e n n DOI: 10.4172/1736-4337.1000231 s e G ISSN: 1736-4337 Theory and Applications Research Article Open Access Algebra, Hyperalgebra and Lie-Santilli Theory Davvaza B1*, Santilli RM2 and Vougiouklis T3 1Department of Mathematics, Yazd University, Yazd, Iran 2Institute for Basic Research, P. O. Box 1577, Palm Harbor, FL 34682, USA 3School of Science of Education, Democritus University of Thrace, 68100 Alexandroupolis, Greece Abstract The theory of hyperstructures can offer to the Lie-Santilli Theory a variety of models to specify the mathematical representation of the related theory. In this paper we focus on the appropriate general hyperstructures, especially on hyperstructures with hyperunits. We define a Lie hyperalgebra over a hyperfield as well as a Jordan hyperalgebra, and we obtain some results in this respect. Finally, by using the concept of fundamental relations we connect hyper algebras to Lie algebras and Lie-Santilli-addmissible algebras. Keywords: Algebra; Hyperring; Hyperfield; Hypervector space; where R=T+W, S=W−T and R,S,R±S are non-singular operators [2- Hyper algebra; Lie hyperalgea; Lie admissible hyperalgebra; 25]. Fundamental relation Algebraic hyperstructures are a natural generalization of the Introduction ordinary algebraic structures which was first initiated by Marty [11]. After the pioneered work of Marty, algebraic hyperstructuires have The structure of the laws in physics is largely based on symmetries. been developed by many researchers. A review of hyperstructures can The objects in Lie theory are fundamental, interesting and innovating be found in studies of Corsini [3,4,7,8,24]. This generalization offers in both mathematics and physics. It has many applications to the a lot of models to express their problems in an algebraic way. Several spectroscopy of molecules, atoms, nuclei and hadrons. The central applications appeared already as in Hadronic Mechanics, Biology, role of Lie algebra in particle physics is well known. A Lie-admissible Conchology, Chemistry, and so on. Davvaz, Santilli and Vougiouklis algebra, introduced by Albert [1], is a (possibly non-associative) algebra studied multi-valued hyperstructures following the apparent existence that becomes a Lie algebra under the bracket [a,b] = ab − ba. Examples in nature of a realization of two-valued hyperstructures with hyperunits include associative algebras, Lie algebras and Okubo algebras. Lie characterized by matter-antimatter systems and their extentions where admissible algebras arise in various topics, including geometry of matter is represented with conventional mathematics and antimatter invariant affine connections on Lie groups and classical and quantum is represented with isodual mathematics [6,9,10]. On the other hand, mechanics. the main tools connecting the class of algebraic hyperstructures For an algebra A over a field F, the commutator algebra A− of A with the classical algebraic structures are the fundamental relations is the anti-commutative algebra with multiplication [a,b] = ab − ba [5,8,22,24]. In this paper, we study the notion of algebra, hyperalgebra defined on the vector spaceA . If A− is a Lie algebra, i.e., satisfies the and their connections by using the concept of fundamental relation. Jacobi identity, then A− is called Lie-admissible. Much of the structure We introduce a special class of Lie hyperalgebra. By this class of Lie theory of Lie-admissible algebras has been carried out initially hyperalgebra, we are able to generalize the concept of Lie-Santilli under additional conditions such as the flexible identity or power- theory to hyperstructure case. associativity. Hyperrings, Hyperfields and Hypervector Spaces Santilli obtained Lie admissible algebras (brackets) from a modified Let H be a non-empty set and :HH× →℘* () H be a hyperoperation, form of Hamilton’s equations with external terms which represent a where ℘* ()H is the family of all non-empty subsets of H. The couple general non-self-adjoint Newtonian system in classical mechanics. In (H,ο) is called a hypergroupoid. For any two non-empty subsets A and 1967, Santilli introduced the product AB= ab B of H and x H, we define a∈∈ Ab, B , Aο{x} = Aοx and {x} (A , B ) = λµα AB− BA= ( AB −+ BA ) β ( AB + BA ), (1) οA = xοA. A hypergroupoid (H,ο) is called a semihypergroup if for all a,b,c in H we ∈have (aοb) ο c = aο (bοc). In addition, if for every a where λ = α + β, µ = α, which is jointly Lie admissible and Jordan H, aHHHa== , then, (H,ο) is called a hypergroup. A non-empty admissible while admitting Lie algebras in their classification. Then, subset K of a semihypergroup (H,ο) is called a sub-semihypergroup∈ he introduced the following infinitesimal and finite generalizations of Heisenberg equations dA i=( A , H )=λµ AH− HA, (2) *Corresponding author: Davvaz B, Department of Mathematics, Yazd University, dt Yazd, Iran, Tel: 989138565019; E-mail: [email protected] † Htµλ i− i tH Htµ i Received August 06, 2014; Accepted October 13, 2015; Published October 19, where At()= U () t A (0)() V t = e A (0) e , Ue= , iλ tH † and H V== e UV I 2015 is the Hamiltonian. In 1978, Santilli introduced the following most general known realization of products that are jointly Lie admissible Citation: Davvaza B, Santilli RM, Vougiouklis T (2015) Algebra, Hyperalgebra and Lie-Santilli Theory. J Generalized Lie Theory Appl 9: 231. doi:10.4172/1736- and Jordan admissible 4337.1000231 (A , B ) = ARB− BSA =( ATB −+ BTA ) { AWB + BWA } Copyright: © 2015 Davvaza B, et al. This is an open-access article distributed =[AB , ]**+ { AB , } (3) under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the = (ATH−+ HTA ) { AWH + HWA}, original author and source are credited. J Generalized Lie Theory Appl Volume 9 • Issue 2 • 1000231 ISSN: 1736-4337 GLTA, an open access journal Citation: Davvaza B, Santilli RM, Vougiouklis T (2015) Algebra, Hyperalgebra and Lie-Santilli Theory. J Generalized Lie Theory Appl 9: 231. doi:10.4172/1736-4337.1000231 Page 2 of 5 if it is a semihypergroup. In other words, a non-empty subset K of a • The relationγ * is the smallest equivalence relation such that the semihypergroup (H,ο) is a sub-semihypergroup if KK⊆ K. We say quotient R/γ* be a ring. that a hypergroup (H,ο) is canonical if Theorem 2.3. [12] The relation γ on every hyperfield is an • It is commutative; equivalence relation and γ =γ*. • It has a scalar identity (also called scalar unit), which means that Remark 1. Let (F,+, ) be a hyperfield. Then, F/γ* is a field. → * there exists e H, for all x H, x ο e= x; If φ :F F/γ is the canonical map, then ωφF = {xF∈ | ( x ) = 0} , where 0 is the zero of the fundamental⋅ field F/γ*. • Every element∈ has a unique∈ inverse, which means that for all x H, there exists a unique x-1 H such that e x ο x-1; Let (R,+, ) be a hyperring, (M, +) be a canonical hypergroup and ∈ -1 there exists an external map • It is reversible, which∈ means that if ∈x y ο z, then z y ο x and ⋅ -1 y x ο z . ⋅: R × M →℘* ( M ), ( a , x ) ax ∈ ∈ ∈ In liteature of Davvaz, there are several types of hyperrings and such that for all a,b R and for all x,y M we have hyperfields [8]. In what follows we shall consider one of the most general types of hyperrings. a( x+ y )= ax ++ ay ,( a∈ b ) x = ax + bx ,( ab ) x∈ = a ( bx ), The triple (R, +, ) is a hyperring if then M is called a hypermodule over R. If we consider a hyperfield F instead of a hyperring R, then M is called a hypervector space. •(R, +) is a canonical⋅ hypergroup; Remark 2. Note that it is possible in a hypervector space one or • (R, ) is a semihypergroup such that xx⋅⋅0=0 =0 for all x R, i.e, more of hyperoperations be ordinary operations. 0 is a bilaterally absorbing element; ⋅ ∈ Example 3. Let F be a field and V be a vector space on F. If S is a • the hyperoperation “." is distributive over the hyperoperation subspace of V, we consider the following external hyperoperation:aο “+", which means that for all x, y, z of R we have: x=ax+S, for all a F and x V. Then, V is a hypervector space. x⋅( y + z )= xy ⋅+⋅ xzndx a ( + y ) ⋅ z = xz ⋅+⋅ yz . Algebra and Hyperalgebra∈ ∈ Example 1. Let R={x,y,z,t} be a set with the following hyperoperations: Definition 3.1. Let (L,+, ) be a hypervector space over the hyperfield +⋅xyz t xyz t (F,+, ). Consider the bracket (commutator) hope: xxy z t xxx x x ⋅ * yyx t z yxy x y [,]:⋅ L× L →℘ ():(,) L xy → [, xy ] z z t{, xz } {,} yt z x x{, xz } {, xz } then L is a Lie hyperalgebra over F if the following axioms are t t z{,} yt {, xz } t x y{, xz } {,} yt satisfied: Then, (F, +, ) is a hyperring. (L1) The bracket hope is bilinear, i.e. [λλ11x++ 2 xy 2 ,]=([,] λ 1 xy 1 λ 2 [ xy 2 ,]), [,xλλ11 y++ 2 y 2 ]=([, λ 1 xy 1 ] λ 2 [, xy 2 ]), We call (R, +,⋅ ) a hyperfield if (R, +, ) is a hyperring and (R−{0}, ) xx, , x ,, yy , y∈ L λλ, ∈ F is a hypergroup. for all 12 12 , 12 ; ⋅ ⋅ ⋅ Example 2. Let F ={x,y} be a set with the following hyperoperations: (L2) 0∈ [,]xx, for all x L; +⋅x y xy (L3) 0∈ ([x ,[ yz , ]] ++ [ y ,[ zx∈ , ]] [ z ,[ xy , ]]) , for all x,y L.