Non-abelian Tensor Product of Leibniz Algebras and an Exact Sequence in Leibniz Homology

Abstract

Let 𝔪 and 𝔫 be two-sided ideals of a Leibniz algebra 𝔤 such that 𝔤 = 𝔪 + 𝔫. The goal of the paper is to achieve the exact sequence Ker(𝔪 ★ 𝔫 + 𝔫 ★ 𝔪 → 𝔤) → HL ₂(𝔤) → HL ₂(𝔤/𝔪) ⊕ HL ₂(𝔤/𝔫) → 𝔪 ∩ 𝔫/ [𝔪,𝔫] → HL ₁(𝔤) → HL ₁(𝔤/𝔪) ⊕ HL ₁(𝔤/𝔫) → 0, where HL _★denotes the Leibniz homology with trivial coefficients of a Leibniz algebra and ★ denotes a non-abelian tensor product of Leibniz algebras.

Key Words:

• Leibniz algebra
• Leibniz homology
• Non-abelian tensor product

Acknowledgments

J. M. Casas was supported by Xunta de Galicia, Grant: PGIDT00PXI37101PR. M. Ladra was supported by Xunta de Galicia, Grant: PGIDT01PXI20702PR.