Hopf Algebras and Galois Theory
Author | : Stephen U. Chase |
Publisher | : Springer |
Total Pages | : 139 |
Release | : 2007-01-05 |
Genre | : Mathematics |
ISBN | : 3540361340 |
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Author | : Stephen U. Chase |
Publisher | : Springer |
Total Pages | : 139 |
Release | : 2007-01-05 |
Genre | : Mathematics |
ISBN | : 3540361340 |
Author | : U. Chase |
Publisher | : |
Total Pages | : |
Release | : 1969-05 |
Genre | : |
ISBN | : 9780387046167 |
Author | : Lindsay N. Childs |
Publisher | : American Mathematical Soc. |
Total Pages | : 311 |
Release | : 2021-11-10 |
Genre | : Education |
ISBN | : 1470465167 |
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 | : Stephen U. Chase |
Publisher | : |
Total Pages | : 144 |
Release | : 2014-01-15 |
Genre | : |
ISBN | : 9783662186909 |
Author | : Stefaan Caenepeel |
Publisher | : Springer Science & Business Media |
Total Pages | : 516 |
Release | : 2002-03-31 |
Genre | : Mathematics |
ISBN | : 9781402003462 |
This volume is devoted to the Brauer group of a commutative ring and related invariants. Part I presents a new self-contained exposition of the Brauer group of a commutative ring. Included is a systematic development of the theory of Grothendieck topologies and étale cohomology, and discussion of topics such as Gabber's theorem and the theory of Taylor's big Brauer group of algebras without a unit. Part II presents a systematic development of the Galois theory of Hopf algebras with special emphasis on the group of Galois objects of a cocommutative Hopf algebra. The development of the theory is carried out in such a way that the connection to the theory of the Brauer group in Part I is made clear. Recent developments are considered and examples are included. The Brauer-Long group of a Hopf algebra over a commutative ring is discussed in Part III. This provides a link between the first two parts of the volume and is the first time this topic has been discussed in a monograph. Audience: Researchers whose work involves group theory. The first two parts, in particular, can be recommended for supplementary, graduate course use.
Author | : George Janelidze, Bodo Pareigis, and Walter Tholen |
Publisher | : American Mathematical Soc. |
Total Pages | : 588 |
Release | : |
Genre | : |
ISBN | : 9780821871478 |
Author | : Lindsay Childs |
Publisher | : American Mathematical Soc. |
Total Pages | : 133 |
Release | : 1998 |
Genre | : Mathematics |
ISBN | : 0821810774 |
This volume gives two new methods for constructing $p$-elementary Hopf algebra orders over the valuation ring $R$ of a local field $K$ containing the $p$-adic rational numbers. One method constructs Hopf orders using isogenies of commutative degree 2 polynomial formal groups of dimension $n$, and is built on a systematic study of such formal group laws. The other method uses an exponential generalization of a 1992 construction of Greither. Both constructions yield Raynaud orders as iterated extensions of rank $p$ Hopf algebras; the exponential method obtains all Raynaud orders whose invariants satisfy a certain $p$-adic condition.
Author | : Frédéric Fauvet |
Publisher | : European Mathematical Society |
Total Pages | : 284 |
Release | : 2009 |
Genre | : Mathematics |
ISBN | : 9783037190739 |
This volume is the outcome of a CIRM Workshop on Renormalization and Galois Theories held in Luminy, France, in March 2006. The subject of this workshop was the interaction and relationship between four currently very active areas: renormalization in quantum field theory (QFT), differential Galois theory, noncommutative geometry, motives and Galois theory. The last decade has seen a burst of new techniques to cope with the various mathematical questions involved in QFT, with notably the development of a Hopf-algebraic approach and insights into the classes of numbers and special functions that systematically appear in the calculations of perturbative QFT (pQFT). The analysis of the ambiguities of resummation of the divergent series of pQFT, an old problem, has been renewed, using recent results on Gevrey asymptotics, generalized Borel summation, Stokes phenomenon and resurgent functions. The purpose of the present book is to highlight, in the context of renormalization, the convergence of these various themes, orchestrated by diverse Galois theories. It contains three lecture courses together with five research articles and will be useful to both researchers and graduate students in mathematics and physics.
Author | : Susan Montgomery |
Publisher | : American Mathematical Soc. |
Total Pages | : 258 |
Release | : 1993-10-28 |
Genre | : Mathematics |
ISBN | : 0821807382 |
The last ten years have seen a number of significant advances in Hopf algebras. The best known is the introduction of quantum groups, which are Hopf algebras that arose in mathematical physics and now have connections to many areas of mathematics. In addition, several conjectures of Kaplansky have been solved, the most striking of which is a kind of Lagrange's theorem for Hopf algebras. Work on actions of Hopf algebras has unified earlier results on group actions, actions of Lie algebras, and graded algebras. This book brings together many of these recent developments from the viewpoint of the algebraic structure of Hopf algebras and their actions and coactions. Quantum groups are treated as an important example, rather than as an end in themselves. The two introductory chapters review definitions and basic facts; otherwise, most of the material has not previously appeared in book form. Providing an accessible introduction to Hopf algebras, this book would make an excellent graduate textbook for a course in Hopf algebras or an introduction to quantum groups.
Author | : George Janelidze |
Publisher | : American Mathematical Soc. |
Total Pages | : 582 |
Release | : 2004 |
Genre | : Mathematics |
ISBN | : 0821832905 |
This volume is based on talks given at the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories held at The Fields Institute for Research in Mathematical Sciences (Toronto, ON, Canada). The meeting brought together researchers working in these interrelated areas. This collection of survey and research papers gives an up-to-date account of the many current connections among Galois theories, Hopf algebras, and semiabeliancategories. The book features articles by leading researchers on a wide range of themes, specifically, abstract Galois theory, Hopf algebras, and categorical structures, in particular quantum categories and higher-dimensional structures. Articles are suitable for graduate students and researchers,specifically those interested in Galois theory and Hopf algebras and their categorical unification.