Some properties of m-isoclinism and ID*-derivations in Filippov algebras

ABOUT THE AUTHORS Aslan Doosti is a PhD student of pure mathematics algebra branch. The doctor Farshid Saeedi is my supervisor and the professor Mohammad Reza R. Moghaddam and Mrs doctor Soodabeh Tajnia are advisor teachers. I passed theoretical courses in the group theory and Lie algebras. Then, at the suggestion of supervisor continued my research work in the concepts isoclinism, m-isoclinism and derivations of (n-Lie) Filippov algebras and provided a new definition for m-isoclinism in Filippov algebras. Farshid Saeedi published numerous papers in the field of the groups theory and Lie algebras, especially about the concept of isoclinism of them. he has been considered the concept of Fillippov algebras in recent years. Also, professor M.R.R. Moghaddam and S. Tajnia research in the field of the groups theory and Lie algebras. We are going to investigate the obtained results in my article for n-ary algebras. Furthermore, we’re going to examine new definitions of derivation in Filippov and other n-ary algebras. PUBLIC INTEREST STATEMENT Algebraic structure is among the most important topics in mathematics. There are various algebraic structures such as groups, rings, Lie algebras, Leibniz algebras, Filippov algebras, etc. For an optimal usage, such structures need to be classified. The group theory is an algebraic structure which has been previously applied to develop and create other structures. There are many concepts in group theory, such as homomorphism, isomorphism, isoclinism, substructures such as subgroups, subalgebras and derivations, which can be applied for classification. Algebraic concepts have many applications in other scientific fields. In particular, groups and Lie algebras are used in physical applications. For example, Lie algebras are applied in quantum physics and the physics of symmetry. Received: 21 October 2016 Accepted: 15 February 2017 First Published: 27 March 2017


Introduction and preliminaries
The notion of (n-Lie) Filippov algebra was introduced in 1985 by Filippov (1985). Let F be a field. A Filippov algebra over F is an F-vector space A, together with n-ary multilinear and skew-symmetric operation [x 1 , … , x n ], which satisfies the following generalized jacobi identity Moghaddam and Mrs doctor Soodabeh Tajnia are advisor teachers. I passed theoretical courses in the group theory and Lie algebras. Then, at the suggestion of supervisor continued my research work in the concepts isoclinism, m-isoclinism and derivations of (n-Lie) Filippov algebras and provided a new definition for m-isoclinism in Filippov algebras. Farshid Saeedi published numerous papers in the field of the groups theory and Lie algebras, especially about the concept of isoclinism of them. he has been considered the concept of Fillippov algebras in recent years. Also, professor M.R.R. Moghaddam and S. Tajnia research in the field of the groups theory and Lie algebras. We are going to investigate the obtained results in my article for n-ary algebras. Furthermore, we're going to examine new definitions of derivation in Filippov and other n-ary algebras.

PUBLIC INTEREST STATEMENT
Algebraic structure is among the most important topics in mathematics. There are various algebraic structures such as groups, rings, Lie algebras, Leibniz algebras, Filippov algebras, etc. For an optimal usage, such structures need to be classified. The group theory is an algebraic structure which has been previously applied to develop and create other structures. There are many concepts in group theory, such as homomorphism, isomorphism, isoclinism, substructures such as subgroups, subalgebras and derivations, which can be applied for classification. Algebraic concepts have many applications in other scientific fields. In particular, groups and Lie algebras are used in physical applications. For example, Lie algebras are applied in quantum physics and the physics of symmetry. Note that for n ≥ 2 and m ≥ 1, The subalgebra A m+1 is generated by elements of the form ]. Thus, for convenience, we put t = (n − 1)m + 1 and use the form [x 1 , x 2 , … , x (n−1) , x t ], where x 1 , x 2 , … , x (n−1) ∈ A and x t ∈ A m . The linear map :A → B is called homomorphism of Filippov algebras, if satisfies Moneyhun (1994), introduced the isoclinism Lie algebras analogue of the concept and contrary to the group theory cases. She showed that when two Lie algebras have the same finite dimensions then the two concept of isoclinism and isomorphism coincide. Salemkar and Mirzaei (2010) introduced the notion of the equivalence relation, m-isoclinism between Lie algebras and obtain some criterion under which Lie algebras are m-isoclinic. In the paper Saeedi and Veisi (2014), introduced the concept of isoclinism in Filippov algebras (To get more information see, Bioch, 1976;Darabi & Saeedi 2017;Hekster, 1986;Kasymov, 1987;Parvaneh & Moghaddam, 2011).
In the following, we introduce the concept of m-isoclinism of Filippov algebras.
Definition Let A and B be two Filippov algebras. Then A and B are m-isoclinic, if there exist isomorphisms where the rule of horizontal maps is ( ) and t = (n − 1)m + 1. In this case the pair ( , ) is called an m-isoclinic from A to B and is denoted by A ∼ m B.
By the above definition, it is easy to show that: • If m = 1 and n = 2, then we have isoclinism of Lie algebra, that is presented by Moneyhun (1994).
• If n = 2 and m is a natural number, then the concept of m-isoclinism of Lie algebras accrued which is due to Salemkar and Mirzaie (2010).
• If m = 1 and n ⩾ 2, then we have the definition of isoclinism of Filippov algebras that is presented by Saeedi and Veisi (2014), for the first time.
In order to state our results, we shall need the following lemma, that proof is the same as for Lie algebra (Moneyhun, 1994).

On ID * m -derivations of Filippov algebras
The notion of derivations and generalized derivations are very important in the study of n-ary algebras and many researchers have written articles about this notion. The concept of -derivations was introduced by Filippov (1998). This notion was developed in Lie, prime Lie and Malcev algebras by Filippov (1999Filippov ( , 2000 and simple, classical Lie and jordan superalgebras were studied by Kaygorodov (2007Kaygorodov ( , 2009Kaygorodov ( , 2010, he described (n + 1)-ary derivations of simple n-ary algebras and generalized derivations algebra of semisimple Filippov algebras over an algebraically closed field of characteristic zero in Kaygorodov (2011Kaygorodov ( , 2014aKaygorodov ( , 2014b. Also, the concept of generalized derivations algebras of Lie algebras was introduced by Leger and Luks (2000), they show that the Quasi-derivation algebra of a Lie algebra can be embedded in to the derivation algebra of a larger Lie algebra. Quasiderivations, derivation of Lie superalgebras, generalized derivations of color Lie algebra, Hom-Lie superalgebras and Ternary derivations were investigated in Shestakov (2012), Shestakov (2014), Leger and Luks (2000), Chen, Ma, and Ni (2013), Zhou and Fan (2016), Zhou , Chen , and Ma (2014) and Zhang and Zhang (2010) In the following, we give a useful lemma.

Lemma 2.2 Let A is a Filippov algebra. Then for some m ≥ 1, ID * m (A) is subalgebra of Der(A).
Proof The proof is the same proof of Lemma 1.1.
In the following, we show that if A and B are two m-isoclinic (n-Lie) Filippov algebras, then ID * m (A) and ID * m (B) are isomorhpic. And we show that is a Lie homomorphism. It is easy to see that is a well-defined linear transformation. Let 1 , 2 ∈ ID * m (A), y ∈ B and −1 (y + Z m (B)) = x + Z m (A). Then

Theorem 2.3 Let A and B be two finite dimensional non-abelian m-isoclinic n-Lie algebras. Then
On the other hand, we have Future work: We are going to investigate the obtained results in this article for n-ary algebras. It seems that we can obtaine same results by defining a new n-bracket and also give the new definition for subalgebras of algebra derivations in n-ary algebras. ( ( 2 (x))) − 2 ( ( 1 (x))) = ( 1 ( 2 (x))) − ( 2 ( 1 (x)))