The Schur multiplier and stem covers of Leibniz n-algebras

Given a free presentation 0 -> R -> F ->(rho )G -> 0 of a Leibniz n-algebra G, the quotient R boolean AND[F, ...,F-n]/[R,F, ...(n-1), F] is known as the Schur multiplier of G. In the article, we construct a four-term exact sequence relating the Schur multiplier of G and G/N, from which we derive some formulas concerning dimensions of the underlying vector spaces of the corresponding Schur multipliers. Additionally, this exact sequence is useful to characterize nilpotency of Leibniz n-algebras. Finally, we characterize stem covers of Leibniz n-algebras, showing their existence in case of finite dimension. We also analyze the interaction between stem covers of Leibniz n-algebras and the Schur multiplier.