Capability of nilpotent Lie algebras of small dimension

TL;DR

This paper investigates the capability of small-dimensional nilpotent Lie algebras, providing a direct computational method for dimensions up to 6 and extending results to higher dimensions using properties of the Schur multiplier.

Contribution

It introduces a direct method to determine capability of nilpotent Lie algebras of dimension ≤6 and generalizes the approach for higher dimensions using the Schur multiplier and exterior center.

Findings

Capability detection for dim ≤6 via nonabelian exterior square

Extension of capability criteria to higher dimensions using Schur multiplier

Identification of capability in generalized Heisenberg algebras

Abstract

Given a nilpotent Lie algebra $L$ of dimension $\le 6$ on an arbitrary field of characteristic $\neq 2$, we show a direct method which allows us to detect the capability of $L$ via computations on the size of its nonabelian exterior square $L \wedge L$. For dimensions higher than $ 6$, we show a result of general nature, based on the evidences of the low dimensional case, focusing on generalized Heisenberg algebras. Indeed we detect the capability of $L \wedge L$ via the size of the Schur multiplier $M(L/Z^\wedge(L))$ of $L/Z^\wedge(L)$, where $Z^\wedge(L)$ denotes the exterior center of $L$.