An upper bound for the nilpotency class of Leibniz algebras

The notion of Leibniz algebras is introduced by Blokh in 1965 as a noncommutative version of Lie algebras and rediscovered by Loday in 1993. A Leibniz algebra is a vector space A over a field F together with a bilinear map [ , ] : A × A A usually called the Leibniz bracket of A, satisfying the Leibniz identity: [x,[y,z]] = [[x,y],z] – [[x,z],y] , x,y,z ∈ A. The classification of nilpotent Leibniz algebras is one of the most important subject in the study of Leibniz algebras. In the present paper, we obtain an upper bound for the nilpotency class of a finitely generated nilpotent Leibniz algebra A in terms of the maximum of the nilpotency classes of maximal subalgebras of A and the minimal number of generators of A.

Throughout the paper, any Leibniz algebra A is considered over a fixed field F, c denotes the maximum of the nilpotency classes of maximal subalgebras of A, d is the minimal number of generators of A and ⌊ ⌋ denotes the integral part. Moreover, we inductively define: A 1=A and A n = [A n-1 ,A], for n ≥ 2. Lemma 1. Let A be a nilpotent Leibniz algebra and M be a maximal subalgebra of A. Then M is a two-sided ideal of A. Lemma 2. Let I be a two-sided ideal of a Leibniz algebra A and {x1,x2, … ,xk} be a subset of A which contains at least n elements of I (n ≤ k). Then [[[x1,x2],x3], … ,xk] ∈ I n . Theorem 3. Let A be a finitely generated nilpotent Leibniz algebra such that d > 1. Then A is nilpotent of class at most ⌊𝑐𝑑/(𝑑 − 1)⌋. Corollary 4. Let A be a finitely generated nilpotent Leibniz algebra and {x1,x2, … ,xd} be a minimal generating set of A with d > 1. Then 𝑐𝑙(𝐴) ≤ 𝑚𝑎𝑥 1≤𝑖≤𝑑 ⌊ 𝑐𝑙(𝑀𝑖) 𝑑 𝑑−1 ⌋, where Mi is the two-sided ideal generated by the set { x1 , …, xi-1 , xi+1 , … ,xd} and cl denotes the nilpotency class. Corollary 5. Let A be a finitely generated nilpotent Leibniz algebra of class k with d > 1. Then c ≤ k ≤ 2c. Corollary 6. Let A be a finitely generated nilpotent Leibniz algebra such that d > c+1. Then A is nilpotent of class c. Corollary 7. Let A be a finitely generated nilpotent Leibniz algebra of class 2c such that d > 1. Then d=2. Proposition 8. There is not any non-Lie Leibniz algebra with at least two generators, whose maximal subalgebras are all abelian.