Algebra, Hyperalgebra and Lie-Santilli Theory

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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. xx y xxx y y{, xy } y x y Definition 3.2. Let A be a hypervector space over∈ a hyperfield F. Then, A is called a hyperalgebra over the hyperfield F if there exists a Then, (F, +, ) is a hyperfield. * mapping ⋅:AA × →℘ () A (images to be denoted by x y for x,y A such A Krasner hyperring is a hyperring such that(R, +) is a canonical that the following conditions hold: ⋅ ⋅ ∈ hypergroup with identity 0 and is an operation such that 0 is a (x+ y )= ⋅ z xz ⋅+⋅ yz and xyz⋅( + )= xyxz ⋅+⋅; bilaterally absorbing element. An exhaustive review updated to 2007 of hyperring theory appears in [8]. ⋅ (cx )⋅ y = c ( x ⋅⋅ y )= x ( cy ); Definition 2.1. Let (R, +, ) be a hypering. We define the relation γ 0⋅⋅yy = 0=0; as follows: for all x,y,z A and c F. ⋅ k xγ y⇔∃ n ∈,(,,) ∃ k k ∈ n and[(,,∃ x  x ) ∈ Ri ,(=1,,)] i n 11n i iki In the above∈ definition,∈ if all hyperoperations are ordinary such that operations, then we have an algebra. n ki xy,∈∑ (∏ xij ). Example 4. Let F be a hyperfield and i=1 j=1 Theorem 2.2. [24, 25] Let (R,+, ) be a hypering γ* be the transitive ab A= { | abcd , , ,∈ F }. closure of γ. cd ⋅ * • γ is a strongly regular relation both on (R, +) and (R, ). We define the following hyperoperations on A: * • The quotient R / γ is a ring. ⋅

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ab11  ab 2 2  ab33 [,]ab− = a⋅−⋅ b b a . (5)   ={ | A cd cd cd − 11  2 2  33 A hyperalgebra A is called Lie-admissible if the hyperalgebra A is ∈+ ∈+ ∈+ ∈+ a3 a 1 ab 2, 3 b 1 b 23 , c 1 cd 2 , 3 d 1 d 2 }, a Lie hyperalgebra.

ab11  ab 2 2  ab33  ⊗ ={ | If A is an associative hyperalgebra, then the hyperproduct (5) cd cd cd 11  2 2  33 coincide with (4) and A− is a Lie hyperalgebra in its more usual form. a∈⋅ aa +⋅ bdb, ∈⋅ ab ++⋅ bd, ∈⋅ ca + dcd ⋅, ∈⋅ cb + dd ⋅}, 3 12 123 12 12312 123 12 12 Thus, the associative hyperalgebras constitute a basic class of Lie- abab  11 22 admissible hyperalgebras. a •  = {  |a2∈ aa 1 , b 2 ∈ ab 1 , c 3 ∈∈ ac 23 , ad 2 }. cd11 cd22  A Jordan algebra is a (non-associative) algebra over a field whose Then A together with the above hyperoperations is a hyperalgebra multiplication satisfies the following axioms: over F. (1) xy = yx (commutative law); Example 5. We can generalize Example 4 to n × n matrices. (2) (xy)(xx) =x(y(xx)) (Jordan identity). A non-empty subset A' of a hyperalgebra A is called a sub hyperalgebra if it is a subhyperspace of A and for all xy, ∈ A' we have Definition 3.6.A Jordan hyperalgebra is a (non-associative) hyperalgebra over a hyperfield whose multiplication satisfies the xy∈ A' . following axioms: In connection with the explicit forms of the hyperproduct let us consider an associative hyperalgebra A, with hyperproduct a b, over a [(J1)] x y = y x (commutative law); hyperfield F. It is possible to construct a new hyperalgebra, denoted by [(J2)] (x⋅ y)(x⋅ x) =x (y (x x)) (Jordan identity). A−, by means of the anti-commutative hyperproduct ⋅ Let A be⋅ an ⋅associative⋅ ⋅ hyperalgebra⋅ over a hyperfield F. It is [ab , ]= a⋅−⋅ b b a = x − y . possible to construct a new hyperalgebra, denoted by A+, by means of x∈⋅ ab (4) y∈⋅ ba the commutative hyperproduct Lemma 3.3. For any non-empty subset S of A, we have 0 S − S.. {ab , }= a⋅+⋅ b b a = x + y . x∈⋅ ab (6) Proof. It is straightforward. ∈ y∈⋅ ba − Proposition 3.4. A is a Lie hyperalgebra. Proposition 3.7. A+ is a Jordan hyperalgebra. ∈ λλ∈ Proof. For all xx,12 , x ,, yy 12 , y L, 12, F ,we have Proof. It is straightforward.

[,]=()()λλ11xxyxxyyxx+ 2 2 λλ11+ 2 2 ⋅−⋅ λλ11 + 2 2 Definition 3.8. Corresponding to any hyperalgebra A with hyperproduct a b it is possible to define, as for A−, a commutative =()(λλ11x⋅+ y 2 x 2 ) ⋅−⋅ yy () λλ11 x −⋅ y ( 2 x 2 ) hyperalgebra A+ which is the same hypervector space as A but with the = ((λ11x )⋅−⋅ yy ( λλ11 x )) + (( 2 x 2 ) ⋅−⋅ yy ( λ2 x 2 )) new hyperproduct⋅ =[λλ11xy ,][+ 2 xy 2 ,], =λλ [,]xy+ [ xy ,], 11 2 2 {,}ab+ = a⋅+⋅ b b a . (7) A + and similarly we obtain [,xλλ11 y++ 2 y 2 ]=([, λ 1 xy 1 ] λ 2 [, xy 2 ]). In this connection the most interesting case occure when A is a (commutative) Jordan hyperalgebra. Now, we prove (L2). Since x x is non-empty, there exists a0 x x. hence, −a −xx. Thus, 0∈a − a ⊆ a − a= xx ⋅−⋅ xx =[ xx , ]. A hyperalgebra A is said to be Jordan admissible if A+ is a 0 00 a∈⋅ xx ⋅ ∈ ⋅ (commutative) Jordan hyperalgebra. It remains∈ to show that (L3) is also satisfied. For, [,[,]]=x yz x⋅−⋅ [,] yz [,] yz x If A is an associative hyperalgebra, then the hyperproduct (7) reduces to (6) and A+ is a special Jordan hyperalgebra. Thus, =()()x⋅ yz ⋅−⋅ zy − yz ⋅−⋅ zy ⋅ x associative hyperalgebras constitute a basis class of Jordan-admissible =.xyz⋅⋅−⋅⋅ xzy − yzx ⋅⋅−⋅⋅ zyx hyperalgebras. Hence, * ([x ,[ yz , ]]+ [ yzx ,[ , ]] + [ z ,[ xy , ]]) = xyz⋅⋅−⋅⋅ xzy − yzx ⋅⋅−⋅⋅ zyx Definition 3.9. The fundamental relation ε is defined in a +()yzx ⋅⋅− yxz ⋅⋅−⋅⋅ zxy +⋅⋅ xzy hyperalgebra as the smallest equivalence relation such that the quotient is an algebra. +()zxy ⋅⋅−⋅⋅−⋅⋅+⋅⋅ zyx xyz yxz =()xyz⋅⋅−⋅⋅ xyz +() xzy ⋅⋅ −⋅⋅ xzy By using strongly regular relations, we can connect hyperalgebras +()zyx ⋅⋅−⋅⋅ zyx +() yzx ⋅⋅− yzx ⋅⋅ to algebras. More exactly, starting with a hyperalgebra and using a +()yxz ⋅⋅−⋅⋅ yxz +() zxy ⋅⋅−⋅⋅ zxy.strongly regular relation, we can construct an algebra structure on the quotient set. An equivalence relation ρ on a hyperalgebra A is By Lemma 3.3, 0 is belong to the right hand of the abobe equality, called right (resp. left) strongly regular if and only if xρy implies that so 0∈ ([x ,[ yz , ]] ++ [ y ,[ zx , ]] [ z ,[ xy , ]]) . (xz++ )(ρ yz ) and (xzαρ )( yz⋅ ) for every z A (resp. (zx++ )(ρ zy ) and (zx⋅⋅ )(ρ zy ) ), and ρ is strongly regular if it is both left and right Definition 3.5. Corresponding to any hyperalgebra A with ∈ hyperproduct a b it is possible to define an anticommutative strongly regular. hyperalgebra A− which is the same hypervector space as A with the new Theorem 3.10. Let A be a hyperalgebra over the hyperfield F. Denote ⋅ hyperproduct by U the set of all finite polynomials of elements of A over F. We define the relation ε on A as follows:

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xε yi f and only if{ x , y }⊆∈ u ,w here u U . (2) By (J2), for all xy, ∈ A, {{xy , },{ xx , }} = { x ,{ y ,{ xx , }}} . Hence, * Then, the ε is the transitive closure of ε and is called the fundamental εε**({{xy , },{ xx , }}) = ({ x ,{ y ,{ xx , }}}), equivalence relation on A. which implies that Proof. The proof is similar to the proof of Theorem 3.1 [18]. εε**(),(),xy   εε ** (),() xx  =  ε * (), x  ε * (), y  εε ** (),() xx  . Remark 3. Note that the relation ε* is a strongly regular relation. This completes the proof. Remark 4. In A− /ε*, the binary operations and external operation are defined in the usual manner: In the same way it is possible to introduce always in terms of the associative product the following bilinear form ε*(x )⊕ ε * ( y ) = ε * ( z ), f or z∈+ εε ** ( x ) ( y ), ε*(x ) ε * ( y ) = ε * ( z ), f or z∈⋅ εε ** ( x ) ( y ), (,)=abλ a⋅+ b (1 − λλ ) b ⋅ a = [,] ab +⋅ b a , (10) γ***(r ) ε ( x ) = ε ( z ), f or all∈ γε** ( r ) ( x ). where λ is a free element belonging to the hyperfieldF , which Theorem 3.11. Let A be an associative hyperalgebra over a hyperfield characterizes the λ-mutations A(λ) of A. Clearly, A(1) is isomorphic F. Then, A− /ε* is a Lie-admissible algebra with the following product: to A. In this connection, a more general bilinear form in terms of the 〈〉εεεεεε******(),()=xy () x () y () y (). x (8) associative hyperproduct is given by Proof. By Definition 3.9 and Theorem 3.10, A /ε* is an ordinary associative algebra. So, it is enough to show that it is a Lie algebra with (,)=abλµα a⋅+ b b ⋅ a= [,] ab + β {,}, ab (11) − the hyperproduce (8). By Proposition 3.4, A is a Lie hyperalgebra with where λ=α+β and µ=β−α are free elements belonging to the hyperfield F, the hyperproduct [a,b] = a b − b a. which constitutes the basic hyperproduct of the (λ,µ)-mutations A(λ,µ) xx,, y∈ A λλ∈ of A. Clearly, A(1,0) is isomorphic to A and A(1,−1) is isomorphic to (1) By (L1), for ⋅ all ⋅ 12 , 12, F , we have A−; A(1,1) is isomorphic to A+ and A(λ,1−λ) is isomorphic to A(λ). [λλ11x++ 2 xy 2 ,]=[,] λ 1 xy 1 λ 2 [ xy 2 ,]. Hence, Let A be an associative hyperalgebra. We can define another ()()=()(),λλ11x+ 2 x 2 ⋅−⋅ yy λλ11 x + 2 x 2 λ 1 xyyx 1 ⋅−⋅+1 λ 2 xyyx 2 ⋅−⋅2 hyperoperation by means an element T which is denoted by [,]⋅⋅T and and so is defined by ** ελλ((x+ x ) ⋅−⋅ yy ( λλ x + x )) = ελ ( ( xyyx⋅−⋅ ) + λ ( xyyx ⋅−⋅ )). * 11 2 2 11 2 2 1 1 1 2 2 2 [⋅⋅ , ] :AA × → PA ( ), T (12) ⋅⋅ ⋅⋅−⋅⋅ This implies that [,]:(,)TTxy [, xy ] = xT y yT x. γλ** ε⊕⊕ γλ * ε * ε * ε * γλ ** ε γλ * ε * (()()()())()()(()()()())11xxyyxx 2 2 11 2 2 Proposition 3.13. The hyperoperation [,]⋅⋅T satisfies the following γλεεεε* * ***+ γλεεεε * * *** =()(()()()())()(()()()()).11x yyx1 2 x 2  yyx2 conditions:

Therefore, (1) 0∈ [,]xxT , for all x A; * * * ** 〈⊕γλ()()()(),()11 εx γλ 2 ε xy 2 ε 〉 (2) 0∈ ([,[,x yz ]]TT ++ [,[, y zx ]]TT [,[, z xy ]])TT , for all x, y T. * ** * * * ∈ =γλ (11 )〈 ε (xy ), ε ( ) 〉⊕ γλ (2 ) 〈 ε ( xy 2 ), ε ( ) 〉 . Proof. (1) Since xT⋅⋅ x is non-empty, there exists a∈⋅⋅ xT x. xy,, y∈ A ∈ 0 Similarly, for all 12 , λλ12, ∈ F , we obtain hence, −a0 ∈− xT ⋅ ⋅ x. ** * * * 〈ε(),()()()()xy γλ11 ε ⊕〉 γλ 2 ε y 2 Thus, 0∈a00 − a ⊆ a − a= xT ⋅⋅−⋅⋅ x xT x=[ xx , ]T . * ** * **  a∈⋅ xT ⋅ x =γλ (1 )〈 ε (xy ), ε ( 12 ) 〉⊕ γλ ( ) 〈 ε ( xy ), ε ( 2 ) 〉 . (2) We have ∈⋅−⋅ (2) By (L2), 0 xx xx, so [,[,]]x yzTT = xT⋅⋅ [,] yzTT − [,] yz ⋅⋅ T x

** **** ⋅⋅ ⋅⋅−⋅⋅ − ⋅⋅−⋅⋅ ⋅⋅ εε(0)= (xxxxxxxx⋅−⋅ ) = εεεε () () () () =()xT yTz zTy() yTz zTy Tx ** =.xTyTz⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅⋅⋅ xTzTy yTzTx zTyTx =〈〉εε (),().xx Hence,

(3) By (L3), we have 0∈ ([x ,[ yz , ]] ++ [ y ,[ zx , ]] [ z ,[ xy , ]]) , for all x, y ([,[,x yz ]]TT++ [,[, y zx ]]TT [,[, z xy ]])TT A. Thus, = xTyTz⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅⋅⋅ xTzTy yTzTx zTyTx

* * ** *** * ** +()yTzTx ⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅⋅⋅+⋅⋅⋅⋅ yTxTz zTxTy xTzTy ∈ ε(0) =〈 ε (x ), 〈 εε ( yz ), ( ) 〉〉 + 〈 εεε ( yzx ), ( ), ( )〉〉 + 〈 ε ( z ), 〈 εε ( xy ), ( ) 〉〉 . +()zTxTy ⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅⋅⋅+⋅⋅⋅⋅ zTyTx xTyTz yTxTz Theorem 3.12. Let A be an associative hyperalgebra over a hyperfield =(xT⋅⋅⋅⋅−⋅⋅⋅⋅ yT z xT yT z)( + xT ⋅⋅⋅⋅−⋅⋅ zT y xT z⋅⋅Ty) F. Then, A+ /ε* is a Jordan-admissible algebra with the following product: +()zTyTx ⋅⋅⋅⋅−⋅⋅⋅⋅ zTyTx +() yTzTx ⋅⋅⋅⋅−⋅⋅⋅⋅ yTzTx εε**(),()=xy ε * () x ε * () y ⊕ ε * () y ε * (). x (9) +()yTxTz ⋅⋅⋅⋅−⋅⋅⋅⋅ yTxTz +() zTxTy ⋅⋅⋅⋅−⋅⋅⋅⋅ zTxTy. Proof. By Definition 3.9 and Theoremt1, A /ε* is an ordinary By Lemma 3.3, 0 is belong to the right hand of the abobe equality, associative algebra. So, it is enough to show that it is a Jordan so 0∈ ([,[,x yz ]]TT ++ [,[, y zx ]]TT [,[, z xy ]])TT . algebra with the hyperproduct (9). By Proposition 3.7, A+ is a Jordan Remark 5. If A is an algebra and [,]⋅⋅T :AA × → A, then we have hyperalgebra with the hyperproduct {a,b} = a b − b a. Lie-Santilli braket. (1) By (J1), for all x,y A,{x,y}={y,x}. So, εε**()=()xy⋅+⋅ yx yx ⋅+⋅ xy which ⋅ ⋅ Definition 3.14. Corresponding to any hyperalgebra A with ** ****** implies that εεεεεεεε()xyyxyxxy ()⊕⊕ ()  ()= () () () (). * ∈ hyperproduct a b it is possible to define a A which is the same Thus, εε**(xy ), ( ) = εε ** ( yx ), ( ) ⋅

Page 5 of 5 hypervector space as A with the new hyperproduct t ()T where ()Tij m× n denotes the transpose of the matrix ij m× n . [,]ab = aT⋅⋅−⋅⋅ b bT a. A (13) References A hyperalgebra A is called Lie-Santilli-admissible if the hyperalgebra 1. Albert AA (1948) Power-associative rings. Trans Amer Math Soc 64: 552-593. * A is a Lie hyperalgebra. 2. Anderson R, Bhalekar AA, Davvaz B, Muktibodh PS, Tangde VM (2012) An introduction to Santilli’s isodual theory of antimatter and the open problem of Corollary 3.15. If A is an associative hyperalgebra, then the detecting antimatter asteroids. Numat Bulletin 6: 1-33. hyperproduct (13) coincide with (12) and A* is a Lie hyperalgebra. 3. Corsini P (1993) Prolegomena of Hypergroup Theory. Aviani Editor, Italy. Theorem 3.16. Let A be an associative hyperalgebra over a hyperfield  * 4. Corsini P, Leoreanu-Fotea V (2003) Applications of Hyperstructure Theory. F. Then, A / ε is a Lie-Santilli-admissible algebra with the following Advances in Mathematics 5. product: 5. Davvaz B (2008) A new view of fundamental relations on hyperrings. A survey: 10th International Congress on Algebraic, Hyperstructures and Applications. 〈〉εεεεεεεε******** (14) (),()=()()()()()().xyT x T y y  T x Brno, Sept 39: 43-55. Proof. By Definition 3.9 and Theorem 3.10, A/ε* is an ordinary 6. Davvaz B (2011) A brief survey of applications of generalized algebraic structures and Santilli-Vougiouklis hyperstructures. Proceedings of the associative algebra. So, it is enough to show that it is a Lie algebra with Third International Conference on Lie-Admissible Treatment of Irreversible * the hyperproduce (4). By corollary 3.15, A is a Lie hyperalgebra with Processes (ICLATIP-3), Kathmandu University, Nepal: 91-110. the hyperproduct [,]ab = aT⋅⋅−⋅⋅ b bT a. T 7. Davvaz B (2013) Polygroup Theory and Related Systems. World Scientific xx,, y∈ Publishing Co. Pte. Ltd., Hackensack, NJ. (1) By (L1), for all 12 , λλ12, ∈ F , we have 8. Davvaze B, Leoreanu-Fotea V (2007) Hyperring Theory and Applications. [λλ11x++ 2 xy 2 ,]=TT λ1 [,] xy 1 λ 2 [ xy 2 ,] T. Hence, International Academic Press, USA. ()λλx+ xTyyT ⋅⋅−⋅⋅ ()=()(λλ x + x λ xTyyTx ⋅⋅−⋅⋅ +λ xTyyTx ⋅⋅−⋅⋅ ), 11 2 2 11 2 2 1 1 1 2 2 2 9. Davvaz B, Santilli RM, Vougiouklis T (2011) Studies of multivalued and so hyperstructures for the characterization of matter-antimatter systems and their extension. Algebras Groups and Geometries 28: 105-116. ελλ**((x+ xTyyT ) ⋅⋅−⋅⋅(λλ x + x )) = ελ ( ( xTyyTx⋅⋅−⋅⋅)( +λ xTyyTx ⋅⋅−⋅⋅ )). 11 2 2 11 2 2 1 1 1 2 2 2 10. Davvaz B, Santilli RM, Vougiouklis T (2013) Multi-Valued Hypermathematics for Characterization of Matter and Antimatter Systems. Journal of Computational This implies that Methods in Sciences and Engineering (JCMSE) 13: 37-50. (()()()())()()γλ** εx⊕ γλ * ε * x ε * Ty ε * 11 2 2 11. Marty F (1934) Sur une generalization de la notion de group. Proceedings of * * ** * * ε()()(()()()())yT ε γλ11 ε x⊕ γλ 2 ε x 2 the 8th Congres des Mathematiciens Scandinave, Swedan 45-49. γλεεεεεε* * ***** =()(()()()()()())11 x T y yT  x1 12. Mirvakili S, Anvariyeh SM, Davvaz B (2008) Transitivity of γ-relation on * * ***** . ⊕ γλεεεεεε()(()()()()()()).22 x T y yT  x2 hyperfields Bull. Math. Soc. Sci. Math. Roumanie 51: 233-243. Therefore, 13. Rezaei AH, Davvaz B, Dehkordi SO (2014) Fundamentals of Γ-Algebra and Γ-dimension. UPB Scientific Bulletin, Series A 76: 11-122. 〈⊕γλ* ε * γλ * ε ** ε 〉 ()()()(),()11x 2 xy 2 T 14. Santilli RM (1968) An introduction to Lie-admissible algebras (*). Nuovo * ** * * * =γλ (11 )〈 ε (xy ), ε ( ) 〉⊕TT γλ (2 ) 〈 ε ( xy 2 ), ε ( ) 〉 . Cimento Suppl 6: 1225-1249. 15. Santilli RM (1969) Dissipativity and Lie-admissible algebras. Meccanica 4: 3-11. Similarly, for all xy,,12 y∈ A, λλ12, ∈ F , we obtain 16. Santilli RM (1984) Applications of Lie-admissible algebras in physics. In: Direct ** * * * 〈ε(),()()()()xy γλ11 ε ⊕〉 γλ 2 ε y 2T universality of the Lie-admissible algebras by Ruggero Maria Santilli (edn.). * ** * ** Hadronic Press, Inc., Nonantum, MA 473. =γλ (1 )〈 ε (xy ), ε ( 12 ) 〉⊕TT γλ ( ) 〈 ε (xy ), ε ( 2 ) 〉 . 17. Santilli RM (1979/80) Initiation of the representation theory of Lie-admissible (2) By (L2), 0∈⋅⋅−⋅⋅xT x xT x, so algebras of operators on Bimodular Hilbert spaces. Proceedings of the Second Workshop on Lie-Admissible Formulations (Sci. Center, Harvard Univ., ** ** *** * εε(0)=(xT⋅⋅−⋅⋅ x xT x )=()()()()()()εεεεεε x T x x  T x Cambridge, Mass, Part B. Hadronic J 3: 440-506. =〈〉εε** (),()xx . T 18. Santilli RM, Vougiouklis T (2013) Lie-admissible hyperalgebra. Journal of Pure (3) By (L3), we have 0∈ ([x ,[ yz , ]] ++ [ y ,[ zx , ]] [ z ,[ xy , ]]) , for all and Applied Mathematics 31: 239-254. xy, ∈ A. Thus, 19. Tallini MS (1991) Hypervector spaces. Proc. Fourth Int. Congress on Algebraic

* * ** *** * ** Hyper- structures and Applications, World Scientific 167-174. ε(0) =〈 ε (x ), 〈 εε ( yz ), ( ) 〉〉TT +〈 εεε ( yzx ), ( ), ( )〉〉TT +〈 ε ( z ), 〈 εε ( xy ), ( ) 〉〉TT . 20. Tsagas G (1996) Studies on the classification of Lie-Santilli algebras. Algebras In some cases we can start from a hyperalgebra which is not a Lie Groups Geom 13: 129-148. hyperalgebra with respect to the Lie braket. However it can be Lie- 21. Tsagas G (1996) Isoaffine connections and Santilli’s iso-Riemannian metrics Santilli hyperalgebra with respect to Lie-Santilli braket. on an iso- manifold. Algebras Groups Geom 13: 149-170.

Example 6. Let F be a hyperfield and 22. Vougiouklis T (2013) The Lie-hyperalgebras and their fundamental relations. Southeast Asian Bulletin of Mathematics 37: 601-614. A= {( aij ) m× n | aF ij ∈ }. 23. Vougiouklis T (1994) Hv -vector spaces Algebraic hyperstructures and Similar to Examples 4 and 5, we can define the hyperoperation  applications (Iasi, 1993). 181-190. • and the external hyperproduct . Note that it is impossible to define 24. Vougiouklis T (1994) Hyperstructures and Their Representations. Hadronic the hyperoperation between two elements of A and so A is not a Lie Press, Inc, Palm Harber, USA. hyperalgebra but it is Lie-Santilli hyperalgebra with respect to ⊗ 25. Vougiouklis T (1991) The fundamental relation in hyperrings. The general [(abij ) m×× n ,( ij ) m n ]() T hyperfield, Proc. Fourth Int. Congress on Algebraic Hyperstructures and ij m× n (15) t Applications (AHA 1990), World Scientific 203-211. =()()()()()(),aTabTaij m×××××× n⊗⊗ ij m n ij m n ij m n ⊗⊗ ij m n ij m n

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