The homology of the lamplighter Lie algebra

TL;DR

This paper investigates the homological properties of the lamplighter Lie algebra's Malcev completion, revealing its uncountably infinite-dimensional homology in each degree, thus advancing understanding of its algebraic structure.

Contribution

It establishes that the homology of the completed lamplighter Lie algebra is uncountably infinite-dimensional in every degree, a novel insight into its algebraic complexity.

Findings

Homology of the completed lamplighter Lie algebra is uncountably infinite-dimensional in each degree.

The associated Lie algebra of the Malcev ${Q}$-completion is the pronilpotent completion of the lamplighter Lie algebra.

The work connects the algebraic structure of the lamplighter group with its Lie algebra homology.

Abstract

We show that the associated Lie algebra of the Malcev $\mathbb{Q}$-completion of the lamplighter group is the pronilpotent completion of the lamplighter Lie algebra. We also prove that the homology of this completed Lie algebra is of uncountable dimension on each degree.