$2$- capability and $2$- nilpotent multiplier of finite dimensional nilpotent Lie algebras

TL;DR

This paper studies the structure and bounds of the 2-nilpotent multiplier in finite-dimensional nilpotent Lie algebras, with specific results on Heisenberg algebras and 2-capability conditions.

Contribution

It characterizes the 2-nilpotent multiplier for Heisenberg Lie algebras, provides inequalities to refine known bounds, and determines 2-capability of Heisenberg algebras.

Findings

Characterization of $ ext{M}^{(2)}(H)$ for Heisenberg Lie algebras.

Inequalities to tighten bounds on $ ext{dim}~ ext{M}^{(2)}(L)$.

H(m) is 2-capable if and only if m=1.

Abstract

In the present context, we investigate to obtain some more results about $2$-nilpotent multiplier $\mathcal{M}^{(2)}(L)$ of a finite dimensional nilpotent Lie algebra $L$. For instance, we characterize the structure of $\mathcal{M}^{(2)}(H)$ when $H$ is a Heisenberg Lie algebra. Moreover, we give some inequalities on $ \mathrm{dim}~ \mathcal{M}^{(2)}(L)$ to reduce a well known upper bound on $2$-nilpotent multiplier as much as possible. Finally, we show that $H(m)$ is 2-capable if and only if m=1.