# The structure, capability and the Schur multiplier of generalized Heisenberg Lie algebras

## TL;DR

This paper investigates the structure, capabilities, and Schur multiplier of generalized Heisenberg Lie algebras, extending known results from group theory to Lie algebra contexts, especially for nilpotent Lie algebras of class two.

## Contribution

It provides new formulas and insights into the Schur multiplier and tensor square of generalized Heisenberg Lie algebras, addressing an open problem in Lie algebra theory.

## Key findings

- Derived formulas for the Schur multiplier of generalized Heisenberg Lie algebras.
- Analyzed the tensor square of certain nilpotent Lie algebras.
- Extended group-theoretic results to Lie algebra settings.

## Abstract

From [Problem 1729, Groups of prime power order, Vol. 3], Berkovich et al. asked to obtain the Schur multiplier and the representation of a group $G$, when $G$ is a special $p$-group minimally generated by $d$ elements and $|G'|=p^{\frac{1}{2}d(d-1)}$. Since there are analogies between groups and Lie algebras, we intend to give an answer to this question similarly for nilpotent Lie algebras. Furthermore, we give some results about the tensor square and the Schur multiplier of some nilpotent Lie algebras of class two.