Estimations of the low dimensional homology of Lie algebras with large abelian ideals

TL;DR

This paper investigates bounds on the second homology (Schur multiplier) of Lie algebras with large abelian ideals, providing new upper bounds and extending previous estimates for nilpotent Lie algebras and their pairs and triples.

Contribution

It introduces new upper bounds for the Schur multipliers of nilpotent Lie algebras, pairs, and triples, refining existing estimates using exact sequences.

Findings

Derived new bounds for the dimension of the Schur multiplier of nilpotent Lie algebras.

Extended bounds to pairs and triples of nilpotent Lie algebras.

Utilized exact sequences to establish these bounds.

Abstract

A Lie algebra $L$ of dimension $n \ge1 $ may be classified, looking for restrictions of the size on its second integral homology Lie algebra $H_2(L,\mathbb{Z})$, denoted by $M(L)$ and often called Schur multiplier of $L$. In case $L$ is nilpotent, we proved that $\mathrm{dim} \ M(L) \leq \frac{1}{2}(n+m-2)(n-m-1)+1$, where $\mathrm{dim} \ L^2=m \ge 1$, and worked on this bound under various perspectives. In the present paper, we estimate the previous bound for $\mathrm{dim} \ M(L) $ with respect to other inequalities of the same nature. Finally, we provide new upper bounds for the Schur multipliers of pairs and triples of nilpotent Lie algebras, by means of certain exact sequences due to Ganea and Stallings in their original form.