# The Schur Lie-Multiplier of Leibinz Algebras

## TL;DR

This paper introduces the Schur Lie-multiplier for Leibniz algebras, providing a new tool to analyze their structure and properties such as Lie-nilpotency, Lie-stem covers, and Lie-capability.

## Contribution

It defines the Schur Lie-multiplier for Leibniz algebras and establishes a four-term exact sequence relating it to quotients, advancing the understanding of Leibniz algebra properties.

## Key findings

- Constructed a four-term exact sequence for Schur Lie-multiplier
- Characterized Lie-nilpotency and Lie-capability using the multiplier
- Provided tools for studying Leibniz algebra extensions

## Abstract

For a free presentation $0 \to R \to F \to G \to 0$ of a Leibniz algebra $G$, the Baer invariant ${\cal M}^{\sf Lie}(G) = \frac{R \cap [F, F]_{Lie}}{[F, R]_{Lie}}$ is called the Schur multiplier of $G$ relative to the Liezation functor or Schur Lie-multiplier. For a two-sided ideal $N$ of a Leibniz algebra $G$, we construct a four-term exact sequence relating the Schur Lie-multiplier of $G$ and $G/N$, which is applied to study and characterize Lie-nilpotency, Lie-stem covers and Lie-capability of Leibniz algebras.