Projection spaces and twisted Lie algebras

TL;DR

This paper introduces the concept of projection spaces, showing their homology can be computed using generalized Lie algebras, and explores conditions for representation stability in classical cases.

Contribution

It defines projection spaces and demonstrates their homology is computed via Chevalley--Eilenberg complexes of generalized Lie algebras, linking topology and algebra.

Findings

Homology of projection spaces is computed by Chevalley--Eilenberg complexes.

Identification of conditions for representation stability.

Application to various configuration spaces and Stiefel manifolds.

Abstract

A projection space is a collection of spaces interrelated by the combinatorics of projection onto tensor factors in a symmetric monoidal background category. Examples include classical configuration spaces, orbit configuration spaces, the graphical configuration spaces of Eastwood--Huggett, the simplicial configuration spaces of Cooper--de Silva--Sazdanovic, the generalized configuration spaces of Petersen, and Stiefel manifolds. We show that, under natural assumptions on the background category, the homology of a projection space is calculated by the Chevalley--Eilenberg complex of a certain generalized Lie algebra. We identify conditions on this Lie algebra implying representation stability in the classical setting of finite sets and injections.