Topological components of spaces of commuting elements in connected nilpotent Lie groups

We study the homotopy type of spaces of commuting elements in connected nilpotent Lie groups, via almost commuting elements in their Lie algebras. We give a necessary and sufficient condition on the fundamental group of such a Lie group $G$ to ensure $Hom (ℤ^{k}, G)$ is path-connected. In particular for the reduced upper unitriangular groups and the reduced generalized Heisenberg groups, $Hom (ℤ^{k}, G)$ is not path-connected, and we compute the homotopy type of its path-connected components in terms of Stiefel manifolds and the maximal torus of $G$ .

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Keywords

commuting elements, connected nilpotent Lie groups

Mathematical Subject Classification

Primary: 22E25

Milestones

Received: 25 June 2024

Revised: 17 April 2025

Accepted: 30 May 2025

Published: 14 February 2026

Authors

Omar Antolín Camarena
Instituto de Matemáticas Universidad Nacional Autónoma de México Mexico City Mexico

Bernardo Villarreal
Departamento de Matemáticas CINVESTAV Mexico City Mexico

Vol. 8, No. 1, 2026

Omar Antolín Camarena and Bernardo Villarreal

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors