Research Article Open Access Davvaza et al., J Generalized Lie Theory Appl 2015, 9:2 DOI: 10.4172/1736-4337.1000231 Volume 9 • Issue 2 • 1000231 J Generalized Lie Theory Appl ISSN: 1736-4337 GLTA, an open access journal Algebra, Hyperalgebra and Lie-Santilli Theory Davvaza B 1 *, Santilli RM 2 and Vougiouklis T 3 1 Department of Mathematics, Yazd University, Yazd, Iran 2 Institute for Basic Research, P. O. Box 1577, Palm Harbor, FL 34682, USA 3 School of Science of Education, Democritus University of Thrace, 68100 Alexandroupolis, Greece *Corresponding author: Davvaz B, Department of Mathematics, Yazd University, Yazd, Iran, Tel: 989138565019; E-mail: davvaz@yazd.ac.ir Received August 06, 2014; Accepted October 13, 2015; Published October 19, 2015 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 Copyright: © 2015 Davvaza B, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 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 defne a Lie hyperalgebra over a hyperfeld 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; Hyperfeld; Hypervector space; Hyper algebra; Lie hyperalgea; Lie admissible hyperalgebra; Fundamental relation Introduction Te structure of the laws in physics is largely based on symmetries. Te objects in Lie theory are fundamental, interesting and innovating in both mathematics and physics. It has many applications to the spectroscopy of molecules, atoms, nuclei and hadrons. Te central role of Lie algebra in particle physics is well known. A Lie-admissible algebra, introduced by Albert [1], is a (possibly non-associative) algebra that becomes a Lie algebra under the bracket [a,b] = ab − ba. Examples include associative algebras, Lie algebras and Okubo algebras. Lie admissible algebras arise in various topics, including geometry of invariant afne connections on Lie groups and classical and quantum mechanics. For an algebra A over a feld F, the commutator algebra A − of A is the anti-commutative algebra with multiplication [a,b] = ab − ba defned on the vector space A. If A − is a Lie algebra, i.e., satisfes the Jacobi identity, then A − is called Lie-admissible. Much of the structure theory of Lie-admissible algebras has been carried out initially under additional conditions such as the fexible identity or power- associativity. Santilli obtained Lie admissible algebras (brackets) from a modifed form of Hamilton’s equations with external terms which represent a general non-self-adjoint Newtonian system in classical mechanics. In 1967, Santilli introduced the product ( , )= = ( ) ( ), AB AB BA AB BA AB BA λ µ α β − − + + (1) where λ = α + β, µ = α, which is jointly Lie admissible and Jordan admissible while admitting Lie algebras in their classifcation. Ten, he introduced the following infnitesimal and fnite generalizations of Heisenberg equations =( , )= , dA i AH AH HA dt λ µ − (2) where † ()= () (0) ()= (0) Ht i i tH At UtA Vt e A e µ λ − , = Ht i U e µ , † = = i tH V e UV I λ and H is the Hamiltonian. In 1978, Santilli introduced the following most general known realization of products that are jointly Lie admissible and Jordan admissible * * ( , ) = =( ) { } =[ , ] {, } =( ) { }, AB ARB BSA ATB BTA AWB BWA AB AB ATH HTA AWH HWA − − + + + − + + (3) where R=T+W, S=W−T and R,S,R±S are non-singular operators [2- 25]. Algebraic hyperstructures are a natural generalization of the ordinary algebraic structures which was frst initiated by Marty [11]. Afer the pioneered work of Marty, algebraic hyperstructuires have been developed by many researchers. A review of hyperstructures can be found in studies of Corsini [3,4,7,8,24]. Tis generalization ofers a lot of models to express their problems in an algebraic way. Several applications appeared already as in Hadronic Mechanics, Biology, Conchology, Chemistry, and so on. Davvaz, Santilli and Vougiouklis studied multi-valued hyperstructures following the apparent existence in nature of a realization of two-valued hyperstructures with hyperunits characterized by matter-antimatter systems and their extentions where matter is represented with conventional mathematics and antimatter is represented with isodual mathematics [6,9,10]. On the other hand, the main tools connecting the class of algebraic hyperstructures with the classical algebraic structures are the fundamental relations [5,8,22,24]. In this paper, we study the notion of algebra, hyperalgebra and their connections by using the concept of fundamental relation. We introduce a special class of Lie hyperalgebra. By this class of Lie hyperalgebra, we are able to generalize the concept of Lie-Santilli theory to hyperstructure case. Hyperrings, Hyperfelds and Hypervector Spaces Let H be a non-empty set and * : ( ) H H H × →℘  be a hyperoperation, where * ( ) H ℘ is the family of all non-empty subsets of H. Te couple (H,ο) is called a hypergroupoid. For any two non-empty subsets A and B of H and x ∈ H, we defne , = a Ab B A B a b ∈ ∈    , Aο{x} = Aοx and {x} 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 ∈ H, = = a H H H a   , then, (H,ο) is called a hypergroup. A non-empty subset K of a semihypergroup (H,ο) is called a sub-semihypergroup Journal of Generalized Lie Theory and Applications G e n e r a l i z e d L i e T h e o r y a n d A p p l i c a t i o n s ISSN: 1736-4337