Unstable algebras over an operad

Abstract

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. For this study we focus on the case of characteristic 2. We define ⋆unstable P-algebras, where P is an operad and ⋆ is a commutative binary operation in P. We then build a functor that takes an unstable module M to the free ⋆-unstable P-algebra generated by M . Under some hypotheses on ⋆ and on M , we identify this unstable algebra as 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 internal product.

Unstable algebras over an operad II

“…Both the Brown-Gitler and the Carlsson modules are endowed with a multiplication which naturally arises when studying their structure. It has been shown that certain of these modules, with their respective multiplication, are in fact free objects in a specified category of (non-associative) algebra [11,17].…”

“…In [17], we introduced a notion of unstable algebra over an operad in characteristic two. For an operad P and a commutative operation ⋆ ∈ P(2), a ⋆-unstable P-algebra is an unstable module A over the mod 2 Steenrod algebra, endowed with an action of P, satisfying the following identity, called instability:…”

“…The main result of [17] shows the existence of an isomorphism of graded P-algebras between the free ⋆-unstable P-algebra generated by an unstable module M , and the free P-algebra generated by the underlying graded vector space of the unstable module ΣΩM , where Σ is the suspension functor and Ω is left adjoint to Σ. One of the conditions for this result to hold is that the operation ⋆ is required to be P-central, which means that it satisfies a certain interchange relation with respect to all other operations in P. This interchange relation can be interpreted as a generalised commutativity relation, which justifies the choice of the term "central".…”

“…G. Powell pointed out that the result in [17] is mainly concerned with F 2 -vector spaces endowed with a Frobenius map, that is, an endomorphism that doubles the degree. In our case, this Frobenius map is the 'top' Steenrod square Sq 0 .…”

“…We recall the construction of the partition operad and we introduce a new family of operads encoding a 'p-level' operation. We define the notion of a central operation of an operad, which generalises the notion of central binary operation of [17].…”