Classification Of Nilpotent Leibniz 3-Algebras

Leibniz algebras were introduced by J.-L.Loday in 1993, then J.-L.Loday, J.M.Casas and T.Pirashvili generalized it to Leibniz n-algebras. In 2006, S.Albeverio, B.A.Omirov and I.S.Rakhimov classified the 4-dimensional nilpotent complex Leibniz algebras. In this thesis, we mainly study the classification of 3-dimensional two-step nilpotent Leibniz 3-algebras.In section 1, we recall some definitions and notations for the theory of Leibniz 3-algebras, such as. definitions of Leibniz 3-algebras, center, nilpotent, strong-nilpotent and two-step nilpotent.In section 2, we prove the subspace [L, L1,L]≠0 if and only if [L,L, L1]≠0 in the 3-dimensional two-step nilpotent Leibniz 3-algebras, and give the classification of 3-dimensional two-step nilpotent Leibniz 3-algebras which the dimension of the derived algebra is one.In section 3, we give the classification of 3-dimensional two-step nilpotent Leibniz 3-algebras which the dimension of the derived algebra is two.In section 4, we give the classification of 3-dimensional strong-nilpotent Leibniz 3-algebras.