On certain three algebras generated by binary algebras
aa r X i v : . [ m a t h . R A ] F e b ON CERTAIN THREE ALGEBRAS GENERATED BY BINARY ALGEBRAS
H. AHMED , , M.A.A. AHMED , , SH.K. SAID HUSAIN , , WITRIANY BASRI Department of Math., Faculty of Science, UPM, Selangor, Malaysia Institute for Mathematical Research (INSPEM), UPM, Serdang, Selangor, Malaysia Depart. of Math., Faculty of Science, Taiz University, Taiz, Yemen
Abstract.
This paper’s central theme is to prove the existence of an n-algebra whose multiplica-tion cannot be expressed employing any binary operation. Furthermore, to prove if two algebras arenot isomorphic, this property does not hold for 3-algebras corresponding to these two algebras. Theproof drives applying some results gotten early applying a new approach for the classification alge-bras problem, introduced recently, which showed great success in solving many classification algebrasproblems. emails: houida [email protected]; mohammed [email protected]; [email protected]; [email protected]
H. Ahmed, et.al. Introduction
In 1969 [11], Kurosh introduced the notion of multilinear operator algebra. It is known that suchalgebraic structures are attractive for their applications to problems of modern mathematical physics.In 1973 [14], Nambu proposed an exciting generalization of classical Hamiltonian mechanics; theNambu bracket is a generalization of the classical Poisson bracket.Indeed, the advance of theoretical physics of quantum mechanics and the discovery of Nambumechanics (see [14]), together with Okubo’s work on the Yang-Baxter equation (see [15]), gave impetusto significant development on triple algebra (3-algebras).Furthermore, Carlsson, Lister, and Loos have studied triple algebra of associative type (see [9, 12,13]). Hestenes provided the typical and founding example of totally associative triple algebra (see[10]).In this article, we give basic definitions and examples related to general n -algebras, and we shallfocus our attention on 3-algebras structures generated by binary algebras presented recently in [1].Then, we introduce the definition of totally associative 3-algebras with examples, which show. Owingto the large size of the matrices involved in our computations of totally associative 3-algebras, wepresent only Mathematica’s results. 2. Preliminaries
Let F be any field and the product A ⊗ B is the Kronecker product which stands for the matrixwith blocks ( a ij B ) , where A = ( a ij ) and B are matrices over F . Definition 2.1.
A vector space A over F with multiplication · : A × A → A given by ( u , v ) u · v such that • ( α u + β v ) · w = α ( u · w ) + β ( v · w ) , • w · ( α u + β v ) = α ( w · u ) + β ( w · v ) , whenever u , v , w ∈ A and α, β ∈ F , is said to be an algebra. Definition 2.2.
Two algebras A and B are called isomorphic if there is an invertible linear map f : A → B such that (2.1) f ( u · A v ) = f ( u ) · B f ( v ) whenever u , v ∈ A . Definition 2.3.
A vector space V over F equipped by a multilinear map f : V × V × ... × V | {z } n − times −→ V issaid to be a n -algebra, that means: • f ( x , x , ..., x i + x ′ i , ..., x n ) = f ( x , x , ..., x i , ..., x n ) + f ( x , x , ..., x ′ i , ..., x n ) • f ( x , x , ..., λx i , ..., x n ) = λf ( x , x , ..., x i , ..., x n ) where ( x , x , ..., x i , ..., x n ) ∈ V. Example 2.4.
Let A = ( V, µ ) be an algebra over a field F : Then multilinear map (2.2) f ( x , x , . . . , x n ) = µ ( x , µ ( x , ..., µ ( x n − , x n ) ... )) | {z } ( n − − times defines an n -algebra structure on V . n certain -algebras generated by binary algebras Classification approach of m -dimensional -algebras Let A be m -dimensional 3-algebra over F and e = ( e , e , ..., e m ) its basis. Then the multilinearmap · is represented by a matrix A = ( A lijk ) ∈ M ( m × m ; F ) as follows(3.1) u · v · w = e A ( u ⊗ v ⊗ w ) , for u = e u, v = e v, w = e w, where u = ( u , u , ..., u m ) T , v = ( v , v , ..., v m ) T and w = ( w , w , ..., w m ) T are column coordinate vectors of u , v , and w , respectively. The matrix A ∈ M ( m × m ; F ) definedabove is called the matrix of structural constants (MSC) of A with respect to the basis e . Further weassume that a basis e is fixed and we do not make a difference between the algebra A and its MSC A .If e ′ = ( e ′ , e ′ , ..., e ′ m ) is another basis of A , e ′ g = e with g ∈ G = GL ( m ; F ), and A ′ is MSC of A with respect to e ′ then it is known that(3.2) A ′ = gA ( g − ) ⊗ is valid (see [8]). Thus, we can reformulate the isomorphism of algebras as follows. Definition 3.1.
Two m -dimensional -algebras A , B over F , given by their matrices of structureconstants A , B , are said to be isomorphic if B = gA ( g − ) ⊗ holds true for some g ∈ GL ( m ; F ) . Further we consider only the case m = 2 then A can be represented by its matrix of structuralconstants (MSC) A = ( γ lijk ) ∈ M (2 × ; F ) where i, j, k, l = 1 , γ lijk ∈ F , as follows: A = (cid:18) γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ (cid:19) (for more information refer to [2]).4. 3 -algebras generated by binary algebras Due to [1] we have the following classification theorems according to
Char ( F ) = 2 , . Theorem 4.1.
Over an algebraically closed field F ( Char ( F ) = 2 and , any non-trivial -dimensionalalgebra is isomorphic to only one of the following algebras listed by their matrices of structure constants: • A ( c ) = (cid:18) α α α + 1 α β − α − α + 1 − α (cid:19) , where c = ( α , α , α , β ) ∈ F , • A ( c ) = (cid:18) α β β − α (cid:19) ≃ (cid:18) α − β β − α (cid:19) , where c = ( α , β , β ) ∈ F , • A ( c ) = (cid:18) β β − (cid:19) , where c = ( β , β ) ∈ F , • A ( c ) = (cid:18) α β − α (cid:19) , where c = ( α , β ) ∈ F , • A ( c ) = (cid:18) α α − − α (cid:19) , where c = α ∈ F , • A ( c ) = (cid:18) α β − α − α (cid:19) ≃ (cid:18) α − β − α − α (cid:19) , where c = ( α , β ) ∈ F , • A ( c ) = (cid:18) β − (cid:19) , where c = β ∈ F , • A ( c ) = (cid:18) α − α − α (cid:19) , where c = α ∈ F , H. Ahmed, et.al. • A = (cid:18) − (cid:19) , • A = (cid:18) − (cid:19) , • A = (cid:18) − (cid:19) , • A = (cid:18) (cid:19) . Example 4.2.
In example 2.4, if V is -dimensional vector space with a fixed basis { e , e } and ( V, f ) is -algebra structure on V. Then f and µ can be expressed by their structure constants as follows: f ( e i , e j , e k ) = γ ijk e + γ ijk e and µ ( e r , e s ) = η rs e + η rs e . Then due to (2.2) we get the system ofequations (4.1) γ ijk = η jk η i + η jk η i γ ijk = η jk η i + η jk η i ) we get = 16 equations for = 8 unknowns (the coefficients η kij ), which cannot be solved in general,except maybe for some very special cases.Indeed, using (4.1), we can find the -algebras corresponding to all algebras presented in [1] under thisprocedure (see Table 1). From the table we can see • (cid:18) − − (cid:19) is a 3-algebra which cannot be expressed by any algebras inthe above theorem and it is not isomorphic to any B i where i = 1 , ..., . • On the other hand, A (1 / , − /
3) and A (1 /
3) are not isomorphic algebras, but from thesetwo non-isomorphic algebras, we get one 3-algebra (cid:18) − (cid:19) . • Also A (1 , −
1) and A (1 ,
1) are not isomorphic algebras but from these two algebras we getone 3-algebra (cid:18) (cid:19) . Totally associative -algebras In this section, we introduce the following definition of totally associative 3-algebra using its MSC.
Definition 5.1. A -algebra A is a totally associative -algebra if (5.1) ( u · v · w ) · x · y = u · ( v · w · x ) · y = u · v · ( w · x · y ) for all u , v , w , x , y ∈ A . Example 5.2.
Let { e , e } be a basis of a -dimensional -algebra A , the multilinear map ” · ” givenby: e · e · e = e , e · e · e = e defines a totally associative -algebra. According to (5.1) and (3.1), we can reformulate Definition 5.1 as follows. n ce r t a i n - a l g e b r a s g e n e r a t e db y b i n a r y a l g e b r a s algebra 3-algebra A ( c ) B ( c ) = (cid:18) α β + α α + α α α − α α β + α α + α α − α − α α α + 2 α − α α + α + 1 α α β + α α β + α − α + β α β + α α − α β − α + α α − α α − α α + α (cid:19) A ( α , β , β ) B ( α , β , β ) = (cid:18) α α β β − α α β + β β β (1 − α ) β β α (1 − α ) 0 0 1 − α (cid:19) A ( β , β ) B ( β , β ) = (cid:18) β β − β β β + β β + β − β − β − β (cid:19) A ( α , β ) B ( α , β ) = (cid:18) α β (1 − α ) β α (1 − α ) 0 0 0 (cid:19) A ( α ) B ( α ) = (cid:18) α α − α − (2 α −
1) (1 − α ) 0 α (1 − α ) 0 0 0 (cid:19) A ( α , β ) B ( α , β ) = (cid:18) α α β − α − α β (1 − α ) − α (1 − α ) β − α − α (cid:19) A ( β ) B ( β ) = (cid:18) β β + 1 0 − β β − − β − (cid:19) A ( α ) B ( α ) = (cid:18) α α − α ) − α (1 − α ) 0 0 0 0 0 (cid:19) A B = (cid:18) − − (cid:19) A B = (cid:18) − (cid:19) A B = (cid:18) − − (cid:19) A trivial Table 1. c = ( α , α , α , β )) H. Ahmed, et.al.
Definition 5.3. A -dimensional -algebra A with multiplication · over a field F is said to be a totallyassociative -algebra if all of the following conditions are met: A ( A ⊗ I ⊗ I − I ⊗ A ⊗ I ) = 0 , (5.2a) A ( A ⊗ I ⊗ I − I ⊗ I ⊗ A ) = 0 , (5.2b) A ( I ⊗ A ⊗ I − I ⊗ I ⊗ A ) = 0 , (5.2c) where I is the identity × matrix. Using a computation program (here, we use Mathematica), it is easy to verify that the 3-algebrasfrom the list in Table (1) satisfying the system (5.2) are:(i) B ( α , β , β ) when • α = 0 , β = 0 , β = 0 , • α = , β = 0 , β = − , • α = , β = 0 , β = . (ii) B ( α , β ) when • α = 0 , β = 0 , • α = , β = − , • α = , β = 0 , • α = , β = , • α = 1 , β = − , • α = 1 , β = 0 , • α = 1 , β = 1 . That means from the above list we get the following totally associative 3-algebras:(i) B (0 , ,
0) = (cid:18) (cid:19) , (ii) B ( , , − ) = (cid:18) −
12 12
14 14 (cid:19) , (iii) B ( , , ) = (cid:18)
12 12
14 14 (cid:19) (iv) B ( , − ) = (cid:18) − (cid:19) , (v) B ( ,
0) = (cid:18) (cid:19) , (vi) B ( , ) = (cid:18)
14 14 (cid:19) , (vii) B (1 , −
1) = B (1 ,
1) = (cid:18) (cid:19) , (viii) B (1 ,
0) = (cid:18) (cid:19) . According to ([5]), over an algebraically closed field, F of characteristic, not 2 , A (cid:18) , , (cid:19) =
12 0 0 10 12 12 0 , n certain -algebras generated by binary algebras (ii) A (1 ,
0) = (cid:18) (cid:19) , (iii) A (cid:18) , (cid:19) =
12 0 0 00 12 12 0 , (iv) A (1 ,
1) = (cid:18) (cid:19) , (v) A (cid:18) , (cid:19) =
12 0 0 00 0 12 0 , (vi) A = (cid:18) (cid:19) . We conclude that there are totally associative 3-algebras, B (0 , , , B ( , , − ) , B ( , − ) , B (1 , − , are generated by non-associative algebras, A (0 , , , A ( , , − ) , A ( , − ) , A (1 , − , respectively. Conclusion
Depending on the approach introduced in [8] and applied in [1], one can study the classification of n -algebras and then study some identities of n -algebras (refer to [2, 3, 4, 5, 6, 7]). Acknowledgement
This research was funded by Grant 01-01-18-2032FR Kementerian Pendidikan Malaysia.