Unstable algebras over an operad II

$\def\P\{\mathcal{P}}$We work over the finite field $\mathbb{F}_q$. We introduce a notion of unstable $\P$-algebra over an operad $\P$. We show that the unstable $\P$-algebra freely generated by an unstable module is itself a free $\P$-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 in terms of free unstable $q$-level algebras.

Keywords

Steenrod algebra, bialgebra, unstable module, operad

2010 Mathematics Subject Classification

17A30, 55S10

Full Text (PDF format)

Received 10 January 2023

Received revised 27 February 2023

Accepted 28 February 2023

Published 21 February 2024