Semisimple Leibniz algebras, their derivations and automorphisms

Abstract

The present paper is devoted to the description of finite-dimensional semisimple Leibniz algebras over complex numbers, their derivations and automorphisms. (2010): 17A32, 17A60, 17B10, 17B20. AMS Subject Classifications

Local Automorphisms on Finite-Dimensional Lie and Leibniz Algebras

“…Let L = S+I be a simple complex Leibniz algebra. In [9] it was proved that an automorphism σ of S can be extended to an automorphism ϕ of L if and only if I ≃ I σ as S-modules. In [9] we also have present an example which shows the existence of automorphism of S which can not be extended to the whole algebra L.…”

“…In particular, a nilpotent Lie algebra L is called filiform if dim L k = n − k − 1 for 1 ≤ k ≤ n − 1. It is known [20] that there exists a basis {e 1 , e 2 , · · · , e n } of L such that (9) [e 1 , e i ] = e i+1 for all i ∈ 2, n − 1.…”

“…From (9) it follows that {e k+2 , · · · , e n } is a basis in L k for all 1 ≤ k ≤ n − 2 and e n ∈ Z(L). Since [L 1 , L n−3 ] ⊆ L n−1 = {0}, it follows that (10) [e i , e n−1 ] = 0 for all i = 3, · · · , n. Define a linear operator Φ on L by (11) Φ (x) = x + αx 2 e n−1 + x 3 e n , x = n k=1…”

The description of the automorphism groups of finite-dimensional cyclic Leibniz algebras

“…Many new results can be found in the survey papers [6][7][8]. In the study of various types of Leibniz algebras, the information about their endo morphisms and derivations is very useful, as shown, for example, in [9,10]. The endomorphisms and derivations of infinite-dimensional cyclic Leibniz algebras were investigated in [11].…”