On the \(\mathrm{mod}p\) Steenrod algebra and the Leibniz-Hopf algebra
type: Item
instance of: scholarly article
title: On the \(\mathrm{mod}p\) Steenrod algebra and the Leibniz-Hopf algebra
zbMATH Open document ID: 1453.55013
author: Neşet Deniz Turgay
published in: Electronic Research Archive
publication date: 2020-07-14T00:00:00Z
review text: For an odd prime \(p\), let \(\mathcal{A}_p\) denote the Bockstein-free part of the \(\mathrm{mod} \ p\) Steenrod algebra (which is generated as an associative algebra by the Steenrod powers \(\mathcal{P}^i\)). Using the Milnor coproduct definition, one gets a Hopf algebra structure for \(\mathcal{A}_p\), and its conjugation map is the subject of the present paper. The author considers the Leibnitz-Hopf algebra and its \(\mathrm{mod} \ p\) reduction, \(\mathcal{F}_p\), defined as the free associative \(\mathbb{Z}/p\) algebra on generators \(S^1, S^2, \ldots\), and carrying a natural Hopf algebra structure (using again a version of the Milnor coproduct). If each \(S^i\) is given the same degree as the corresponding \(\mathcal{P}^i\), by considering the Adem relations one gets a quotient map \(\pi : \mathcal{F}_p \rightarrow \mathcal{A}_p\). The point of this paper is to use this map, together with the known formula for the conjugation on \(\mathcal{F}_p\) (appearing here at the beginning of Section 3), to determine the conjugation of some admissible elements of \(\mathcal{A}_p\). The method is exemplified in Section 3, and is most successful for those that are images of elements of \(\mathcal{F}_p\) whose conjugation becomes simpler once one applies the Adem relations. The last section of this work gives a new, combinatorial proof of a result on the coarsening operation in the \(\mathrm{mod} \ 2\) dual Leibnitz-Hopf algebra, \(\mathcal{F}_2^*\), and presents an additional result on the conjugation invariants in \(\mathcal{F}_2^*\) using its algebra structure.
reviewed by: Rui Miguel Saramago
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 57T05
zbMATH DE Number: 7220340
zbMATH Keywords: Hopf algebra
zbMATH Keywords: Leibniz-Hopf algebra
zbMATH Keywords: Steenrod algebra
Wikidata QID: Q122979703
describes a project that uses: Steenrod
describes a project that uses: Leibniz-Hopf
describes a project that uses: Maple
describes a project that uses: SageMath
MaRDI profile type: Publication
full work available at URL: era.2020050
OpenAlex ID: W3034681866
cites work: Q5589091
cites work: Monomial bases in the Steenrod algebra
cites work: On the Anti-Automorphism of the Steenrod Algebra
cites work: Quasi-Hopf Algebras
cites work: On the Adem relations
cites work: The intersection of the admissible basis and the Milnor basis of the Steenrod algebra
cites work: The Steenrod Algebra and Other Copolynomial Hopf Algebras
cites work: SOME HOPF ALGEBRAS OF WORDS
cites work: Conjugation Invariants in the Mod 2 Dual Leibniz–Hopf Algebra
cites work: Conjugation invariants in the Leibniz-Hopf algebra.
cites work: On conjugation invariants in the dual Steenrod algebra
cites work: Higher conjugation cohomology in commutative Hopf algebras
cites work: The Antiautomorphism of the Steenrod Algebra
cites work: Q5749433
cites work: On posets and Hopf algebras
cites work: On monomial bases in the mod \(p\) Steenrod algebra
cites work: Graphical calculus of Hopf crossed modules
cites work: Generalized overlapping shuffle algebras.
cites work: The algebra of quasi-symmetric functions is free over the integers
cites work: Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions
cites work: Symmetric functions, noncommutative symmetric functions and quasisymmetric functions. II.
cites work: Explicit polynomial generators for the ring of quasisymmetric functions over the integers.
cites work: On conjugation in the mod-\(p\) Steenrod algebra
cites work: Monomial Bases in the Mod-p Steenrod Algebra
cites work: Duality between quasi-symmetric functions and the Solomon descent algebra
cites work: The Steenrod algebra and its dual
cites work: On the structure of Hopf algebras
cites work: Change of basis, monomial relations, and \(P_ t^ s\) bases for the Steenrod algebra
cites work: Cohomologie modulo 2 des complexes d'Eilenberg-MacLane
cites work: Conjugation and Excess in the Steenrod Algebra
cites work: Cohomology Operations (AM-50)
cites work: Identities for Conjugation in the Steenrod Algebra
cites work: Quelques propriétés globales des variétés différentiables
cites work: Q3466489
cites work: A REMARK ON THE CONJUGATION IN THE STEENROD ALGEBRA
cites work: An alternative approach to the Adem relations in the mod 2 Steenrod algebra
cites work: Q5371090
cites work: Invariants under decomposition of the conjugation in the mod 2 dual Leibniz-Hopf Algebra
cites work: The mod 2 dual Steenrod algebra as a subalgebra of the mod 2 dual Leibniz-Hopf algebra
cites work: The nilpotence height of $Sq^{2^n}$
cites work: The nilpotence height of P p n
cites work: Generators and relations for the Steenrod algebra
cites work: A note on bases and relations in the Steenrod algebra
DOI: 10.3934/ERA.2020050
description: scientific article; zbMATH DE number 7220340