The c-Nilpotent Shur Lie-Multiplier of Leibniz Algebras

TL;DR

This paper introduces the c-nilpotent Schur Lie-multiplier for Leibniz algebras, providing formulas, characterizations, and existence results to deepen understanding of their structure and nilpotency properties.

Contribution

It defines the c-nilpotent Schur Lie-multiplier for Leibniz algebras and establishes formulas, existence theorems, and characterizations related to Lie-nilpotency and c-Lie-stem covers.

Findings

Derived exact sequences and dimension formulas for the c-nilpotent Schur Lie-multiplier.

Proved existence of c-Lie-stem covers for finite dimensional Leibniz algebras.

Characterized c-Lie-capability using c-Lie-characteristic ideal and c-nilpotent Schur Lie-multiplier.

Abstract

We introduce the notion of c-nilpotent Schur Lie-multiplier of Leibniz algebras. We obtain exact sequences and formulas of the dimensions of the underlying vector spaces relating the c-nilpotent Schur Lie-multiplier of a Leibniz algebra Q and its quotient by a two-sided ideal. These tools are used to characterize Lie-nilpotency and c-Lie-stem covers of Leibniz algebras. We prove the existence of c-Lie-stem covers for finite dimensional Leibniz algebras and the non existence of c-covering on certain Lie-nilpotent Leibniz algebras with non trivial c-nilpotent Schur Lie-multiplier, and we provide characterizations of c-Lie-capability of Leibniz algebras by means of both their c-Lie-characteristic ideal and c-nilpotent Schur Lie-multiplier.