The Steenrod Algebra and Other Copolynomial Hopf Algebras

Crossley, Martin D.

doi:10.1112/s0024609300007128

Cited by 10 publications

(13 citation statements)

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“…Hazewinkel proved the conjecture in [16]. For another approach on related matters from a topological perspective see [8]. Here is our statement of these results.…”

Section: A Proof Of the Ditters Conjecturementioning

confidence: 76%

Quasisymmetric functions from a topological point of view

Baker¹,

Richter²

2008

MATH. SCAND.

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It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions Symm. We offer the cohomology of the space CP ∞ as a topological model for the ring of quasisymmetric functions QSymm. We exploit standard results from topology to shed light on some of the algebraic properties of QSymm. In particular, we reprove the Ditters conjecture. We investigate a product on CP ∞ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU . The canonical Thom spectrum over CP ∞ is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.

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“…Hazewinkel proved the conjecture in [16]. For another approach on related matters from a topological perspective see [8]. Here is our statement of these results.…”

Section: A Proof Of the Ditters Conjecturementioning

confidence: 76%

Quasisymmetric functions from a topological point of view

Baker¹,

Richter²

2008

MATH. SCAND.

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“…Ditters himself discovered that the proof was incomplete and several attempts to patch this have been given and, in most cases, found wanting. In [1] we proved that the algebra was polynomial over Z/p (for any prime p) and, although the integral statement can be deduced from this, the details were not given in that paper. Around the same time Hazewinkel introduced the distinction between the 'Ditters conjecture' (that the algebra was polynomial) and the 'strong Ditters conjecture' (that it was the polynomial algebra on the ESL words) and gave a complete proof of the Ditters conjecture [5].…”

Section: Dramatis Personae -The Operationsmentioning

confidence: 99%

“…The algebra B, denoted M by Hazewinkel, is more familiar as the ring of quasi-symmetric functions with the outer coproduct, as defined by [6]. It is known to topologists as the cohomology of CP ∞ , and the dual algebra B * , referred to by Hazewinkel as the 'Leibniz-Hopf algebra' is isomorphic to the Solomon Descent algebra [6,9], and is the ring of 'noncommutative symmetric functions' of [4], and the integral lift of the algebra F of [1].…”

Section: Dramatis Personae -The Operationsmentioning

confidence: 99%

“…To be more specific, let us henceforth assume that S is the set N of non-negative integers, with the element n ∈ S having degree n. Then for C we could take the Hopf algebra B and, using the first few terms of PQ B (t) given in [1], we see that, when p = 2, the number of generators (i.e. the dimension of the module of indecomposables) in each degree in A ⊗ Z/2 is given by the following At the time of writing, Neil Sloane's online handbook of integer sequences had no information on this sequence.…”

Section: Dramatis Personae -The Operationsmentioning

confidence: 99%

“…As noted in [1], for each prime p, the algebra B * ⊗ Z/p has a particularly important quotient A p , the mod p Steenrod algebra (to be precise, for p odd, this is the Bockstein-free part of the Steenrod algebra, but we will use the term Steenrod algebra in both cases). Consequently, the dual Steenrod algebra is a sub Hopf algebra of B ⊗ Z/p, which leads to one quick proof of the fact that the dual Steenrod algebra is polynomial, a result originally due to Milnor [7].…”

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confidence: 99%

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Some Hopf Algebras of Words

Crossley¹

2006

Glasgow Math. J.

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Abstract. A number of integral Hopf algebras have been studied that have, as their underlying modules, the free Z-module generated by finite words in a certain alphabet. For example, the tensor algebra, the rings of quasisymmetric functions and of noncommutative symmetric functions, the Solomon descent algebra, the MalvenutoReutenauer algebra and the homology and cohomology of CP ∞ are all of this type. Some of these are known to be isomorphic or dual to each other, some are known only to be rationally isomorphic, some have been stated in the literature to be isomorphic when they are only rationally isomorphic.This paper is, in part, an attempt to find order in this chaos of word Hopf algebras. We consider three multiplications on such modules, and their dual comultiplications, and clarify which of these operations can be combined to obtain Hopf structures. We discuss when the results are isomorphic, integrally or rationally, and study the resulting structures. We are not attempting a classification of Hopf algebras of words, merely an organization of some of the Hopf algebras of this type that have been studied in the literature.2000 Mathematics Subject Classification. 16W30, 57T05, 05E05, 55S10.1. The basic set-up. Given a set S, thought of as a collection of letters, we can form the free monoid WS consisting of finite words in S, the monoidal operation being composition. Here it is assumed that a word has positive length; we let WS denote the unital monoid obtained by adjoining to WS a unique 'empty word' of length 0 (which will give the unit and counit in the algebras and coalgebras we consider). Then we can take the free abelian groups ZWS and ZWS generated by these monoids, the elements of these groups being Z-linear combinations of words. Of course ZWS = ZWS ⊕ Z.We shall restrict ourselves to the graded setting, and so we assume that S is given a grading in which every element has positive degree and only finitely many elements have the same degree. The degree of a word is defined to be the sum of the degrees of the letters that form it and so ZWS becomes graded and of finite type; i.e., the degree n part, ZWS n , of ZWS, has finite rank. We can then form the graded dual (ZWS) * = n (ZWS n ) * , and when we speak of duality we shall always mean this graded duality. Each ZWS n has an obvious basis consisting of the words of degree n, and we give (ZWS n ) * the dual basis, collating all of these to provide a basis for (ZWS) * . Of course, the elements of this dual basis are indexed by words, giving a Z-linear isomorphism between ZWS and (ZWS) * . A Hopf algebra structure consists of a multiplication on ZWS, and a compatible comultiplication on ZWS, the antipode being given automatically since ZWS is graded and connected. Moreover, a comultiplication can be considered as the dual of a multiplication on (ZWS) * which, by at https://www.cambridge.org/core/terms. https://doi

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