Characterizing nilpotent Leibniz algebras by a new bound on their second homologies

Let L be a finite dimensional nilpotent Leibniz algebra such that dim⁡(L)=n and dim⁡(L2)=m≠0. In this paper, we prove dim⁡(HL2(L))≤(n+m−2)(n−m)−m+2, where HL2(L) is the second Leibniz homology of L. As a consequence, for a non-abelian nilpotent Leibniz algebra L, we find that s(L)=(n−1)2+1−dim⁡(HL2(L))≥0. Furthermore, we determine all finite dimensional nilpotent Leibniz algebras with s(L) less than or equal to three.