A classification of polynomial algebras as modules over the Steenrod algebra

Duflot, Jeanne; Kuhn, Nicholas J.; Winstead, Mark

doi:10.1007/bf02565838

Cited by 3 publications

(14 citation statements)

References 6 publications

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“…The last part of this article is devoted to the generalisation of the results obtained by Campbell and Selick in [6], and by Duflot, Kuhn and Winstead in [14].…”

mentioning

confidence: 76%

“…Since several examples of unstable modules of interest come equipped with operations that are not associative, we argue that polynomial algebras do not hold a particular place in a general study of unstable modules, and we extend the constructions and results obtained in [6] and [14] to free objects in different categories of algebras. We use the formalism of operads which allows us to offer a common framework for many algebraic structures.…”

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confidence: 93%

“…A natural question that arises is to find conditions in which an algebra can be equipped with a compatible unstable action of the Steenrod algebra. Such actions on polynomial algebras are studied by Campbell and Selick [6], and by Duflot, Kuhn and Winstead [14]. In [6], the authors use techniques borrowed from representation theory of finite fields to show that the action of A on the mod p cohomology of simple abelian p-groups can be twisted, yielding an isomorphism of unstable modules between a certain polynomial algebra and itself with two different A-actions.…”

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confidence: 99%

“…In [14], the authors use a notion of higher derivations to prove that any linear transformation of the linear subspace of a polynomial algebra yields an unstable action of A on that polynomial algebra. Following the example of [6], they also give conditions for two such linear transformations to induce an equivalent structure of unstable module.…”

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confidence: 99%

“…Following [14], we then show how to extend a linear transformation of a vector space to an action of A ′ on the free algebra generated by this vector space. Adapting the techniques from [14] leads us to define a notion of higher operadic derivation which generalises both the notion of higher derivation due to [1] and the notion of operadic derivation [25]. We prove the following: Theorem E (Theorem 6.2.3).…”

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confidence: 99%

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Unstable algebras over an operad II

Ikonicoff¹

2020

Preprint

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We introduce a notion of unstable algebra over an operad. We show that the unstable algebra freely generated by an unstable module is itself a free algebra under suitable conditions. This arises as a particular case of a result on graded algebras endowed with a Frobenius map. We use our result to obtain a new construction of the unstable modules studied by Carlsson and Brown-Gitler which takes into account their internal product. Finally, we generalise results due to Campbell-Selick and Duflot-Kuhn-Winstead, which deal with twisted actions on unstable algebras.

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“…The last part of this article is devoted to the generalisation of the results obtained by Campbell and Selick in [6], and by Duflot, Kuhn and Winstead in [14].…”

mentioning

confidence: 76%

mentioning

confidence: 93%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Unstable algebras over an operad II

Ikonicoff¹

2020

Preprint

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Realizing coalgebras over the Steenrod algebra

Blanc

2001

Topology

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Unstable algebras over an operad

Ikonicoff

2021

Homology, Homotopy and Applications

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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.

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A classification of polynomial algebras as modules over the Steenrod algebra

Cited by 3 publications

References 6 publications

Unstable algebras over an operad II

Unstable algebras over an operad II

Realizing coalgebras over the Steenrod algebra

Unstable algebras over an operad

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