Taming Wild Extensions Hopf Algebras And Local Galois Module Theory

Author : Lindsay Childs
Publisher : American Mathematical Soc.
Page : 225 pages
File Size : 49,6 Mb
Release : 2000
Category : Field extensions (Mathematics).
ISBN : 9780821821312

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Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory by Lindsay Childs Pdf

This book studies Hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. The results, not previously published in book form, show that Hopf algebras play a natural role in local Galois module theory. Included in this work are expositions of short exact sequences of Hopf algebras; Hopf Galois structures on separable field extensions; a generalization of Noether's theorem on the Galois module structure of tamely ramified extensions of local fields to wild extensions acted on by Hopf algebras; connections between tameness and being Galois for algebras acted on by a Hopf algebra; constructions by Larson and Greither of Hopf orders over valuation rings; ramification criteria of Byott and Greither for the associated order of the valuation ring of an extension of local fields to be Hopf order; the Galois module structure of wildly ramified cyclic extensions of local fields of degree p and p2; and Kummer theory of formal groups. Beyond a general background in graduate-level algebra, some chapters assume an acquaintance with some algebraic number theory. From there, this exposition serves as an excellent resource and motivation for further work in the field.

Author : Lindsay N. Childs,Cornelius Greither,Kevin P. Keating,Alan Koch,Timothy Kohl,Paul J. Truman,Robert G. Underwood
Publisher : American Mathematical Soc.
Page : 311 pages
File Size : 41,5 Mb
Release : 2021-11-10
Category : Education
ISBN : 9781470465162

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Hopf Algebras and Galois Module Theory by Lindsay N. Childs,Cornelius Greither,Kevin P. Keating,Alan Koch,Timothy Kohl,Paul J. Truman,Robert G. Underwood Pdf

Hopf algebras have been shown to play a natural role in studying questions of integral module structure in extensions of local or global fields. This book surveys the state of the art in Hopf-Galois theory and Hopf-Galois module theory and can be viewed as a sequel to the first author's book, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, which was published in 2000. The book is divided into two parts. Part I is more algebraic and focuses on Hopf-Galois structures on Galois field extensions, as well as the connection between this topic and the theory of skew braces. Part II is more number theoretical and studies the application of Hopf algebras to questions of integral module structure in extensions of local or global fields. Graduate students and researchers with a general background in graduate-level algebra, algebraic number theory, and some familiarity with Hopf algebras will appreciate the overview of the current state of this exciting area and the suggestions for numerous avenues for further research and investigation.

Author : Jeffrey Bergen,Susan Montgomery
Publisher : CRC Press
Page : 344 pages
File Size : 42,5 Mb
Release : 2023-08-18
Category : Mathematics
ISBN : 9781000938890

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Advances in Hopf Algebras by Jeffrey Bergen,Susan Montgomery Pdf

"This remarkable reference covers topics such as quantum groups, Hopf Galois theory, actions and coactions of Hopf algebras, smash and crossed products, and the structure of cosemisimple Hopf algebras. "

Author : Robert G. Underwood
Publisher : Springer
Page : 164 pages
File Size : 42,5 Mb
Release : 2015-06-10
Category : Mathematics
ISBN : 9783319189918

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Fundamentals of Hopf Algebras by Robert G. Underwood Pdf

This text aims to provide graduate students with a self-contained introduction to topics that are at the forefront of modern algebra, namely, coalgebras, bialgebras and Hopf algebras. The last chapter (Chapter 4) discusses several applications of Hopf algebras, some of which are further developed in the author’s 2011 publication, An Introduction to Hopf Algebras. The book may be used as the main text or as a supplementary text for a graduate algebra course. Prerequisites for this text include standard material on groups, rings, modules, algebraic extension fields, finite fields and linearly recursive sequences. The book consists of four chapters. Chapter 1 introduces algebras and coalgebras over a field K; Chapter 2 treats bialgebras; Chapter 3 discusses Hopf algebras and Chapter 4 consists of three applications of Hopf algebras. Each chapter begins with a short overview and ends with a collection of exercises which are designed to review and reinforce the material. Exercises range from straightforward applications of the theory to problems that are devised to challenge the reader. Questions for further study are provided after selected exercises. Most proofs are given in detail, though a few proofs are omitted since they are beyond the scope of this book.

Author : Michiel Hazewinkel,Nadezhda Mikhaĭlovna Gubareni,Vladimir V. Kirichenko
Publisher : American Mathematical Soc.
Page : 425 pages
File Size : 49,9 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821852620

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Algebras, Rings and Modules by Michiel Hazewinkel,Nadezhda Mikhaĭlovna Gubareni,Vladimir V. Kirichenko Pdf

Presenting an introduction to the theory of Hopf algebras, the authors also discuss some important aspects of the theory of Lie algebras. This book includes a chapters on the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups, the Hopf algebras of the nonsymmetric and quasisymmetric functions, and the Hopf algebra of permutations.

Author : Fernando Chamizo,Jordi Guàrdia, Antonio Rojas-León,José María Tornero
Publisher : American Mathematical Soc.
Page : 244 pages
File Size : 47,8 Mb
Release : 2015-09-28
Category : Number theory
ISBN : 9780821898581

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Trends in Number Theory by Fernando Chamizo,Jordi Guàrdia, Antonio Rojas-León,José María Tornero Pdf

This volume contains the proceedings of the Fifth Spanish Meeting on Number Theory, held from July 8-12, 2013, at the Universidad de Sevilla, Sevilla, Spain. The articles contained in this book give a panoramic vision of the current research in number theory, both in Spain and abroad. Some of the topics covered in this volume are classical algebraic number theory, arithmetic geometry, and analytic number theory. This book is published in cooperation with Real Sociedad Matemática Española (RSME).

Author : Robert G. Underwood
Publisher : Springer Science & Business Media
Page : 283 pages
File Size : 47,9 Mb
Release : 2011-08-30
Category : Mathematics
ISBN : 9780387727653

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An Introduction to Hopf Algebras by Robert G. Underwood Pdf

Only book on Hopf algebras aimed at advanced undergraduates

Author : Stephen U. Chase,Moss E. Sweedler
Publisher : Springer
Page : 139 pages
File Size : 50,5 Mb
Release : 2007-01-05
Category : Mathematics
ISBN : 9783540361343

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Hopf Algebras and Galois Theory by Stephen U. Chase,Moss E. Sweedler Pdf

Author : George Janelidze, Bodo Pareigis, and Walter Tholen
Publisher : American Mathematical Soc.
Page : 588 pages
File Size : 50,8 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 0821871471

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Galois Theory, Hopf Algebras, and Semiabelian Categories by George Janelidze, Bodo Pareigis, and Walter Tholen Pdf

Author : Jörg Feldvoss,Lauren Grimley,Drew Lewis,Andrei Pavelescu,Cornelius Pillen
Publisher : Springer
Page : 322 pages
File Size : 54,5 Mb
Release : 2019-02-27
Category : Mathematics
ISBN : 9783030115210

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Advances in Algebra by Jörg Feldvoss,Lauren Grimley,Drew Lewis,Andrei Pavelescu,Cornelius Pillen Pdf

This proceedings volume covers a range of research topics in algebra from the Southern Regional Algebra Conference (SRAC) that took place in March 2017. Presenting theory as well as computational methods, featured survey articles and research papers focus on ongoing research in algebraic geometry, ring theory, group theory, and associative algebras. Topics include algebraic groups, combinatorial commutative algebra, computational methods for representations of groups and algebras, group theory, Hopf-Galois theory, hypergroups, Lie superalgebras, matrix analysis, spherical and algebraic spaces, and tropical algebraic geometry. Since 1988, SRAC has been an important event for the algebra research community in the Gulf Coast Region and surrounding states, building a strong network of algebraists that fosters collaboration in research and education. This volume is suitable for graduate students and researchers interested in recent findings in computational and theoretical methods in algebra and representation theory.

Author : Robert G. Underwood
Publisher : Springer Science & Business Media
Page : 283 pages
File Size : 54,6 Mb
Release : 2011-08-28
Category : Mathematics
ISBN : 9780387727660

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An Introduction to Hopf Algebras by Robert G. Underwood Pdf

Only book on Hopf algebras aimed at advanced undergraduates

Author : John Rognes
Publisher : American Mathematical Soc.
Page : 154 pages
File Size : 45,6 Mb
Release : 2008
Category : Commutative algebra
ISBN : 9780821840764

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Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups by John Rognes Pdf

The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.

Author : Tomasz Brzezinski,Robert Wisbauer
Publisher : Cambridge University Press
Page : 492 pages
File Size : 44,6 Mb
Release : 2003-09-15
Category : Mathematics
ISBN : 0521539315

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Corings and Comodules by Tomasz Brzezinski,Robert Wisbauer Pdf

This is the first extensive treatment of the theory of corings and their comodules. In the first part, the module-theoretic aspects of coalgebras over commutative rings are described. Corings are then defined as coalgebras over non-commutative rings. Topics covered include module-theoretic aspects of corings, such as the relation of comodules to special subcategories of the category of modules (sigma-type categories), connections between corings and extensions of rings, properties of new examples of corings associated to entwining structures, generalisations of bialgebras such as bialgebroids and weak bialgebras, and the appearance of corings in non-commutative geometry.

Author : Jordi Guàrdia
Publisher : Springer Nature
Page : 457 pages
File Size : 51,7 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 9783031519598

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New Frontiers in Number Theory and Applications by Jordi Guàrdia Pdf

Author : Florin Felix Nichita
Publisher : MDPI
Page : 239 pages
File Size : 51,8 Mb
Release : 2019-01-31
Category : Mathematics
ISBN : 9783038973249

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Hopf Algebras, Quantum Groups and Yang-Baxter Equations by Florin Felix Nichita Pdf

This book is a printed edition of the Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" that was published in Axioms