Hopf Ring Research Articles

Correction to the article Hopf ring structure on the mod p cohomology of symmetric groups

Correction to the article Hopf ring structure on the mod p cohomology of symmetric groups

Open versus Interpenetrated: Switchable Supramolecular Trajectories in Mechanosynthesis of a Halogen-Bonded Borromean Network

Summary Precise control over topologically intricate molecular architectures remains an open challenge for chemists due to their inherent structural complexity. We report a simple solvent-free strategy to selectively prepare two multi-component supramolecular crystalline systems based on the halogen bond, endowed with either an unknot topology or a Borromean-type entanglement. Real-time in situ monitoring of the solvent-free mechanochemical synthesis of these three-component halogen-bonded ionic co-crystals reveals that the choice of milling conditions leads to a switch in the supramolecular reaction trajectory, resulting in the selective formation of an open halogen-bonded network or a halogen-bonded Borromean-type assembly.

Mod-two cohomology rings of alternating groups

Abstract We calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox–Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.

Jacobson radicals of invariants under Hopf algebra actions

In this paper, we examine the relationship between the Jacobson radicals of rings [Formula: see text] and their invariants [Formula: see text] under the actions of finite dimensional Hopf algebras [Formula: see text]. For large classes of Hopf algebras and rings, we determine when [Formula: see text] or [Formula: see text], for some [Formula: see text].

The spectrum for commutative complex K–theory

We study commutative complex $K$-theory, a generalised cohomology theory built from spaces of ordered commuting tuples in the unitary groups. We show that the spectrum for commutative complex $K$-theory is stably equivalent to the $ku$-group ring of $BU(1)$ and thus obtain a splitting of its representing space $B_{com}U$ as a product of all the terms in the Whitehead tower for $BU$, $B_{com}U\simeq BU\times BU\langle 4\rangle \times BU\langle 6\rangle \times \dots .$ As a consequence of the spectrum level identification we obtain the ring of coefficients for this theory. Using the rational Hopf ring for $B_{com}U$ we describe the relationship of our results with a previous computation of the rational cohomology algebra of $B_{com}U$. This gives an essentially complete description of the space $B_{com}U$ introduced by A. Adem and J. G\'omez.

Hopf ring structure on the mod p cohomology of symmetric groups

We describe a Hopf ring structure on ⊕ n≥0H∗(Σn; ℤp), discovered by Strickland and Turner, where Σn is the symmetric group of n objects and p is an odd prime. We also describe an additive basis on which the cup product is explicitly determined, compute the restriction to modular invariants and determine the action of the Steenrod algebra on our Hopf ring generators. For p = 2 this was achieved in work of Giusti, Salvatore and Sinha, of which this work is an extension.

Ideals of bounded rank symmetric tensors are generated in bounded degree

Over a field of characteristic zero, we prove that for each r, there exists a constant C(r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C(r). The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated. We also prove a similar statement for partial flag varieties and, in fact, arbitrary projective schemes, and we also get multi-graded versions of these results.

Modular coinvariants and the modphomology ofQSk

We use modular invariant theory to establish a complete set of relations of the mod $p$ homology of $\{QS^k\}_{k\geq0}$, for $p$ odd, as a ring object in the category of coalgebras (also known as a coalgebraic ring or a Hopf ring). We also describe the action of the mod $p$ Dyer-Lashof algebra as well as the mod $p$ Steenrod algebra on the coalgebraic ring.

The mod 2 Hopf ring for connective Morava K-theory

This paper examines the mod 2 homology of the spaces in the Omega-spectrum for connective Morava K-theory, i.e., the mod 2 Hopf ring for connective Morava K-theory. A natural set of generators for this Hopf ring arising from the homology and homotopy of the connective Morava K-theory spectrum is calculated and the non-trivial circle product relations among the generators arising from homology and homotopy are determined. This Hopf ring calculation is accomplished using Dieudonne ring theory and Adams spectral sequences for the connective Morava K-theory of Brown–Gitler spectra.

The connective realK–theory of Brown–Gitler spectra

We calculate the connective real K-theory homology of the mod 2 Brown-Gitler spectra. We use this calculation and the theory of Dieudonne rings and Hopf rings to determine the mod 2 homology of the spaces in the connective -spectrum for topological real K-theory. 55T25; 55P42 Zp . He showed that the Dieudonne functor D. / from Hopf rings to Dieudonne rings was symmetric monoidal, thereby establishing an equivalence of categories and, consequently, an isomorphism between any Hopf ring H.E / and its associated Dieudonne ring D.H.E //. Building on his earlier work with Lannes and Morel (11), Goerss also showed there is a surjective map from the E -homology of Brown-Gitler spectra E.B. // to the Dieudonne ring D.H.E // that is periodically an isomorphism. When the mod p (co)homology of the spectrum E is known, the E -homology of the Brown-Gitler spectra E.B. // can be calculated via an Adams spectral sequence. Thus, it is often possible to calculate the Hopf ring H.E / via the Adams spectral sequence for E.B. //. Calculating the Hopf ring H.E / using an Adams spectral sequence for E.B. // is remarkable for several reasons. First, this method for calculating the homology of the spaces E is done using only the (co)homology of the spectrum E and the Brown-Gitler spectra B. / as input to the Adams spectral sequence. Second, this approach can be used even when the spaces E have not been identified in terms of already-known spaces. Third, unlike the bar spectral sequence, which computes H.Ek/ inductively on k (ie, one space Ek at a time), this method computes Hn.E / inductively on n (ie, across all spaces E at once), and as a result it does well

Schubert calculus and the Hopf algebra structures of exceptional Lie groups

Let G be an exceptional Lie group with a maximal torus T. Based on common properties in the Schubert presentation of the cohomology ring H*(G/T;F_{p}) DZ1, and concrete expressions of generalized Weyl invariants for G over F_{p}, we obtain a unified approach to the structure of H*(G;F_{p}) as a Hopf algebra over the Steenrod algebra A_{p}. The results has been applied in Du2 to determine the near--Hopf ring structure on the integral cohomology of all exceptional Lie groups.

A topological method for the classification of entanglements in crystal networksA preliminary account of this work was presented at the workshop `Topological dynamics in physics and biology' held in Pisa, 12–13 July 2011.

A rigorous method is proposed to describe and classify the topology of entanglements between periodic networks if the links are of the Hopf type. The catenation pattern is unambiguously identified by a net of barycentres of catenating rings with edges corresponding to the Hopf links; this net is called the Hopf ring net. The Hopf ring net approach is compared with other methods of characterizing entanglements; a number of applications of this approach to various kinds of entanglement (interpenetration, polycatenation and self-catenation) both in modelled network arrays and in coordination networks are considered.

The mod-2 cohomology rings of symmetric groups

We compute the mod-2 cohomology of the collection of all symmetric groups as a Hopf ring, where the second product is the transfer product of Strickland and Turner. We first give examples of related Hopf rings from invariant theory and representation theory. In addition to a Hopf ring presentation, we give geometric cocycle representatives and explicitly determine the structure as an algebra over the Steenrod algebra. All calculations are explicit, with an additive basis which has a clean graphical representation. We also briefly develop related Hopf ring structures on rings of symmetric invariants and end with a generating set consisting of Stiefel-Whitney classes of regular representations v2. Added new results on varieties which represent the cocycles, a graphical representation of the additive basis, and on the Steenrod algebra action. v3. Included a full treatment of invariant theoretic Hopf rings, refined the definition of representing varieties, and corrected and clarified references.

On the chemical synthesis of new topological structures

The construction of chemical species with topological properties is an area of increasing interest. Numerous such structures have been synthesized in recent years, particularly from DNA, which is a natural synthon for these molecules. Recently, higher-order topological structures have been introduced. These hyper-structures consist of combinations of simple structures, such as a Brunnian Link of Brunnian Links or a Hopf Ring of Hopf Rings, or combinations thereof. In this article, we discuss the possibilities of constructing these hyper-structures with real molecules, particularly emphasizing the tools that result from the double helical structure of DNA.

INDUCED HOPF CORING STRUCTURES

Hopf corings are dened in this work as coring objects in the category of algebras over a commutative ring R. Using the Dieudonn<TEX>$\'{e}$</TEX> equivalences from [7] and [19], one can associate coring structures built from the Hopf algebra <TEX>$F_p[x_0,x_1,{\ldots}]$</TEX>, p a prime, with Hopf ring structures with same underlying connected Hopf algebra. We have that <TEX>$F_p[x_0,x_1,{\ldots}]$</TEX> coring structures classify thus Hopf ring structures for a given Hopf algebra. These methods are applied to dene new ring products in the Hopf algebras underlying known Hopf rings that come from connective Morava <TEX>${\kappa}$</TEX>-theory.

Dieudonné Module Structures for Ungraded and Periodically Graded Hopf Rings

Dieudonne theory provides functors between categories of Hopf algebras and categories of modules over rings. These functors define equivalences of such theories, and thus allow us to represent some Hopf algebras by modules over specific rings, which in some cases can ease calculations one needs to perform in the context of Hopf algebras. The theory can be extended to accommodate Hopf rings, which are Hopf algebras with additional structure. This paper reviews the construction of such equivalences of categories in the case of graded connected Hopf algebras (following Ravenel (1975) and Schoeller (Manuscripta Math 3, 133–155, 1970)) and Hopf rings (following Goerss (1999)); of ungraded connected Hopf algebras (from Bousfield (Math Z 223, 483–519, 1996)); and of periodically graded connected Hopf algebras (from Sadofsky and Wilson (1998)). We also present an alternative proof of Goerss’s equivalence in the graded connected Hopf ring case (Goerss 1999), and complete the picture by constructing the functors for ungraded and periodically graded Hopf rings. Finally, we analyze the unconnected case, focusing on Hopf algebras and rings that are either group-like in degree zero or geometric-like.

The hunting of the Hopf ring

We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, $E*(–)$. Our description is as a graded and completed version of a Tall-Wraith monoid. The $E*$-cohomology of a space $X$ is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.

The ring spectrum $P(n)$ for the prime 2

This paper is a companion to the Boardman‐Wilson paper on the ring spectrum P(n). When the prime is 2, this spectrum is not commutative, which introduces several complications. Here, we supply the necessary details of the relevant Hopf algebroids and Hopf ring for this case.

K(n)-Torsion-Free H-Spaces and P(n)-Cohomology

AbstractThe H-space that represents Brown–Peterson cohomology BPk(–) was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum P(n), by constructing idempotent operations in P(n)–cohomology P(n)k(–) in the style of Boardman–Johnson–Wilson; this relies heavily on the Ravenel–Wilson determination of the relevant Hopf ring. The resulting (i – 1)-connected H-spaces Yi have free connective Morava K-homology k(n)*(Yi), and may be built from the spaces in the Ω-spectrum for k(n) using only vn-torsion invariants.We also extend Quillen's theorem on complex cobordism to show that for any space X, the P(n)*-module P(n)*(X) is generated by elements of P(n)i(X) for i ≥ 0. This result is essential for the work of Ravenel–Wilson–Yagita, which in many cases allows one to compute BP–cohomology from Morava K-theory.

On the Hopf ring for [formula omitted

Kriz and Hu construct a real Johnson–Wilson spectrum, ER ( n ) , which is 2 n + 2 ( 2 n − 1 ) periodic. ER ( 1 ) is just KO ( 2 ) . We do two things in this paper. First, we compute the homology of the 2 n − 1 spaces ER ( n ) ̲ 2 n + 2 k in the Omega spectrum for ER ( n ) . It turns out the double of these Hopf algebras gives the homology Hopf algebras for the even spaces for E ( n ) . As a byproduct of this we get the homology of the zeroth spaces for the Omega spectrum for real complex cobordism and real Brown–Peterson cohomology. The second result is to compute the homology Hopf ring for all 48 spaces in the Omega spectrum for ER ( 2 ) . This turns out to be generated by very few elements.