Unstable algebras over an operad II

TL;DR

This paper develops a theory of unstable algebras over operads in various characteristics, introducing q-level operads to connect with known unstable modules and establishing conditions for free algebra generation.

Contribution

It introduces a general notion of unstable algebras over operads and identifies free unstable algebras via q-level operads, linking to prior unstable module studies.

Findings

Unstable algebra freely generated by an unstable module is itself free under certain conditions.

Introduction of q-level operads to classify unstable modules.

Connection of unstable modules to free unstable q-level algebras.

Abstract

We introduce a notion of unstable algebra over an operad in general characteristic. We show that the unstable algebra freely generated by an unstable module is itself a free algebra under suitable conditions. We introduce a family of '$q$-level' operads which allows us to identify unstable modules studied by Brown--Gitler, Miller and Carlsson to free unstable $q$-level algebras.