Unstable algebras over an operad II

Sacha Ikonicoff

Published 2020-11-25Version 1

The aim of this article is to define and study a notion of unstable algebra over an operad that generalises the classical notion of unstable algebra over the Steenrod algebra in characteristic $p>2$. We first study algebras over an operad in the category of graded vector space with Frobenius map. Under suitable conditions, we show that the free $\mathcal{P}$-algebra with Frobenius map compatible to the action of $\mathcal P$ over a graded vector space $V$ with free Frobenius map is itself a free $\mathcal{P}$-algebra generated by $V$. We then define $\star$-unstable $\mathcal{P}$-algebras over the Steenrod algebra, where $\mathcal{P}$ is an operad and $\star$ is a completely symmetric $p$-ary operation in $\mathcal{P}$. Under some hypotheses on $\star$ and on the unstable module $M$, we identify the free $\star$-unstable $\mathcal{P}$-algebra generated by $M$ as a free $\mathcal{P}$-algebra. We show how to use our main theorem to obtain a new construction of the unstable modules studied by Carlsson, and Brown and Gitler, that takes into account their internal product. Finally, we generalise a result due to Campbell and Selick which shows that we can twist the action of the Steenrod algebra on the free unstable algebra generated by a direct sum of unstable modules without modifying the underlying unstable module structure, and we obtain a decomposition for some of our new constructions which involve the Carlsson modules.