*-Unstable P-algebras

Sacha Ikonicoff

Published 2019-12-17Version 1

The aim of this article is to define and study a notion of unstable algebras over an operad that generalises the classical notion of unstable algebras over the Steenrod algebra. For this study we focus on the case of the characteristic 2. We define $\star$-unstable $P$-algebras, where $P$ is an operad and $\star$ is a commutative binary operation in $P$. We then build a functor that takes an unstable module $M$ to the free $\star$-unstable $P$-algebra generated by $M$. Under some hypotheses on $\star$ and on $M$, we identify this unstable algebra to a free $P$-algebra. Finally, we give some examples of this result, and we show how to use our main theorem to obtain a new construction of the unstable modules studied by Carlsson, Brown-Gitler, and Campbell-Selick, that takes into account their inner product.