Bogomolov Multiplier of Lie algebras

TL;DR

This paper studies the Bogomolov multiplier of finite-dimensional Lie algebras, providing a cohomological characterization and addressing questions about its invariance and unboundedness.

Contribution

It offers a Hopf-Type formula linking the Bogomolov multiplier to second cohomology and answers key questions on its invariance and unboundedness.

Findings

Established a subgroup of $H^2(L, \

Confirmed invariance of the Bogomolov multiplier under isoclinism.

Constructed examples with unbounded Bogomolov multipliers.

Abstract

In the work of Rostami et al., the Bogomolov multiplier of a Lie algebra $L$ over a field $\Omega$ is defined as a particular factor of a subalgebra of the exterior product $L \wedge L$. If $L$ is finite dimensional, we identify this object as a certain subgroup of the second cohomology group $H^2(L, \Omega)$ by deriving a Hopf-Type formula. As an application, we affirmatively answer two questions posed by Kunyavskii regarding the invariance of the Bogomolov multiplier under isoclinism of Lie algebras and the existence of a family of Lie algebras with Bogomolov multipliers of unbounded dimension.