On The Simplest Filiform N-Lie Algebras

An n—Lie algebra is a natural generalization of the concept of a Lie algebra to the case, where the fundamental multiplication operation is n—ary, n ≥ 2 (When n = 2, the definition agrees with the usual definition of a Lie algebra).In this thesis, the concept of the simplest filiform Lie algebra is introduced, which is a natural generalization of the concept of simplest filiform Lie algebra. And then, the derivation algebra Der(A) of such an n—Lie algebra A is determined. And it is proved that if for the base field F, with the characteristic p is p = 0 or p > m — n, and n > 2, Der(A) is an unsolvable complete Lie algebra. And the holomorph h(A) = Der(A) (?) A for such an n—Lie algebra A, has a solvable complete Lie algebra, if p = 0 or p > m — n.At last, the automorphism group Aut(A) of the simplest filiform n-Lie algebra A is determined, and it is proved that it is unsolvable and centerless, if the characteristic of the basis field F is zero. When the characteristic of F is nonzero, the center of Aut(A) is nonzero, and also be determined, if | F | ≥ m.