Download or read book Galois Theory Hopf Algebras and Semiabelian Categories written by George Janelidze and published by American Mathematical Soc.. This book was released on 2004 with total page 582 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Galois Theory Hopf Algebras and Semiabelian Categories written by George Janelidze, Bodo Pareigis, and Walter Tholen and published by American Mathematical Soc.. This book was released on with total page 588 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Brauer Groups Hopf Algebras and Galois Theory written by Stefaan Caenepeel and published by Springer Science & Business Media. This book was released on 2002-03-31 with total page 516 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Hopf Algebras and Galois Theory written by Stephen U. Chase and published by Springer. This book was released on 2007-01-05 with total page 139 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Taming Wild Extensions Hopf Algebras and Local Galois Module Theory written by Lindsay Childs and published by American Mathematical Soc.. This book was released on 2000 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Hopf Algebras and Galois Theory written by Stephen U. Chase and published by . This book was released on 2014-01-15 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Hopf Algebras and Galois Theory Two written by U. Chase and published by . This book was released on 1969-05 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Hopf Algebras Tensor Categories and Related Topics written by Nicolás Andruskiewitsch and published by American Mathematical Soc.. This book was released on 2021-07-06 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles highlight the latest advances and further research directions in a variety of subjects related to tensor categories and Hopf algebras. Primary topics discussed in the text include the classification of Hopf algebras, structures and actions of Hopf algebras, algebraic supergroups, representations of quantum groups, quasi-quantum groups, algebras in tensor categories, and the construction method of fusion categories.

Download or read book Category Theory And Applications A Textbook For Beginners Second Edition written by Marco Grandis and published by World Scientific. This book was released on 2021-03-05 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a better understanding of their roots.This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers the basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications.A reader should have some elementary knowledge of these three subjects, or at least two of them, in order to be able to follow the main examples, appreciate the unifying power of the categorical approach, and discover the subterranean links brought to light and formalised by this perspective.Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications in Algebra and Topology, with a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields.In this second edition, the book has been entirely reviewed, adding many applications and exercises. All non-obvious exercises have now a solution (or a reference, in the case of an advanced topic); solutions are now collected in the last chapter.

Download or read book Handbook of Algebra written by M. Hazewinkel and published by Elsevier. This book was released on 2009-07-08 with total page 592 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it is worthwhile to pursue the quest. In addition to the primary information given in the Handbook, there are references to relevant articles, books or lecture notes to help the reader. An excellent index has been included which is extensive and not limited to definitions, theorems etc. The Handbook of Algebra will publish articles as they are received and thus the reader will find in this third volume articles from twelve different sections. The advantages of this scheme are two-fold: accepted articles will be published quickly and the outline of the Handbook can be allowed to evolve as the various volumes are published. A particularly important function of the Handbook is to provide professional mathematicians working in an area other than their own with sufficient information on the topic in question if and when it is needed. - Thorough and practical source of information - Provides in-depth coverage of new topics in algebra - Includes references to relevant articles, books and lecture notes

Download or read book Galois and Cleft Monoidal Cowreaths Applications written by D. Bulacu and published by American Mathematical Soc.. This book was released on 2021-07-21 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: We introduce (pre-)Galois and cleft monoidal cowreaths. Generalizing a result of Schneider, to any pre-Galois cowreath we associate a pair of adjoint functors L R and give necessary and sufficient conditions for the adjunction to be an equivalence of categories. Inspired by the work of Doi we also give sufficient conditions for L R to be an equivalence, and consequently conditions under which a fundamental structure theorem for entwined modules over monoidal cowreaths holds. We show that a cowreath is cleft if and only if it is Galois and has the normal basis property; this generalizes a result concerning Hopf cleft extensions due to Doi and Takeuchi. Furthermore, we show that the cleft cowreaths are in a one to one correspondence with what we call cleft wreaths. The latter are wreaths in the sense of Lack and Street, equipped with two additional morphisms satisfying some compatibility relations. Note that, in general, the algebras defined by cleft wreaths cannot be identified to (generalized) crossed product algebras, as they were defined by Doi and Takeuchi, and Blattner, Cohen and Montgomery. This becomes more transparent when we apply our theory to cowreaths defined by actions and coactions of a quasi-Hopf algebra, monoidal entwining structures and ν-Doi-Hopf structures, respectively. In particular, we obtain that some constructions of Brzezi´nski and Schauenburg produce examples of cleft wreaths, and therefore of cleft cowreaths, too.

Download or read book Mal cev Protomodular Homological and Semi Abelian Categories written by Francis Borceux and published by Springer Science & Business Media. This book was released on 2004-02-29 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory. The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment. The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics.

Download or read book Associative and Non Associative Algebras and Applications written by Mercedes Siles Molina and published by Springer Nature. This book was released on 2020-01-02 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gathers together selected contributions presented at the 3rd Moroccan Andalusian Meeting on Algebras and their Applications, held in Chefchaouen, Morocco, April 12-14, 2018, and which reflects the mathematical collaboration between south European and north African countries, mainly France, Spain, Morocco, Tunisia and Senegal. The book is divided in three parts and features contributions from the following fields: algebraic and analytic methods in associative and non-associative structures; homological and categorical methods in algebra; and history of mathematics. Covering topics such as rings and algebras, representation theory, number theory, operator algebras, category theory, group theory and information theory, it opens up new avenues of study for graduate students and young researchers. The findings presented also appeal to anyone interested in the fields of algebra and mathematical analysis.

Download or read book Hopf Algebras and Their Generalizations from a Category Theoretical Point of View written by Gabriella Böhm and published by Springer. This book was released on 2018-11-01 with total page 165 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes provide a self-contained introduction to a wide range of generalizations of Hopf algebras. Multiplication of their modules is described by replacing the category of vector spaces with more general monoidal categories, thereby extending the range of applications. Since Sweedler's work in the 1960s, Hopf algebras have earned a noble place in the garden of mathematical structures. Their use is well accepted in fundamental areas such as algebraic geometry, representation theory, algebraic topology, and combinatorics. Now, similar to having moved from groups to groupoids, it is becoming clear that generalizations of Hopf algebras must also be considered. This book offers a unified description of Hopf algebras and their generalizations from a category theoretical point of view. The author applies the theory of liftings to Eilenberg–Moore categories to translate the axioms of each considered variant of a bialgebra (or Hopf algebra) to a bimonad (or Hopf monad) structure on a suitable functor. Covered structures include bialgebroids over arbitrary algebras, in particular weak bialgebras, and bimonoids in duoidal categories, such as bialgebras over commutative rings, semi-Hopf group algebras, small categories, and categories enriched in coalgebras. Graduate students and researchers in algebra and category theory will find this book particularly useful. Including a wide range of illustrative examples, numerous exercises, and completely worked solutions, it is suitable for self-study.

Download or read book Higher Dimensional Categories From Double To Multiple Categories written by Grandis Marco and published by World Scientific. This book was released on 2019-09-09 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of higher dimensional categories has mostly been developed in the globular form of 2-categories, n-categories, omega-categories and their weak versions. Here we study a different form: double categories, n-tuple categories and multiple categories, with their weak and lax versions.We want to show the advantages of this form for the theory of adjunctions and limits. Furthermore, this form is much simpler in higher dimension, starting with dimension three where weak 3-categories (also called tricategories) are already quite complicated, much more than weak or lax triple categories.This book can be used as a textbook for graduate and postgraduate studies, and as a basis for research. Notions are presented in a 'concrete' way, with examples and exercises; the latter are endowed with a solution or hints. Part I, devoted to double categories, starts at basic category theory and is kept at a relatively simple level. Part II, on multiple categories, can be used independently by a reader acquainted with 2-dimensional categories.

Download or read book Homological Algebra written by Marco Grandis and published by World Scientific. This book was released on 2012 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book we want to explore aspects of coherence in homological algebra, that already appear in the classical situation of abelian groups or abelian categories. Lattices of subobjects are shown to play an important role in the study of homological systems, from simple chain complexes to all the structures that give rise to spectral sequences. A parallel role is played by semigroups of endorelations. These links rest on the fact that many such systems, but not all of them, live in distributive sublattices of the modular lattices of subobjects of the system. The property of distributivity allows one to work with induced morphisms in an automatically consistent way, as we prove in a 'Coherence Theorem for homological algebra'. (On the contrary, a 'non-distributive' homological structure like the bifiltered chain complex can easily lead to inconsistency, if one explores the interaction of its two spectral sequences farther than it is normally done.) The same property of distributivity also permits representations of homological structures by means of sets and lattices of subsets, yielding a precise foundation for the heuristic tool of Zeeman diagrams as universal models of spectral sequences. We thus establish an effective method of working with spectral sequences, called 'crossword chasing', that can often replace the usual complicated algebraic tools and be of much help to readers that want to apply spectral sequences in any field.

Download or read book Homological Algebra written by Marco Grandis and published by World Scientific. This book was released on 2013-01-11 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: We propose here a study of ‘semiexact’ and ‘homological' categories as a basis for a generalised homological algebra. Our aim is to extend the homological notions to deeply non-abelian situations, where satellites and spectral sequences can still be studied. This is a sequel of a book on ‘Homological Algebra, The interplay of homology with distributive lattices and orthodox semigroups’, published by the same Editor, but can be read independently of the latter. The previous book develops homological algebra in p-exact categories, i.e. exact categories in the sense of Puppe and Mitchell — a moderate generalisation of abelian categories that is nevertheless crucial for a theory of ‘coherence’ and ‘universal models’ of (even abelian) homological algebra. The main motivation of the present, much wider extension is that the exact sequences or spectral sequences produced by unstable homotopy theory cannot be dealt with in the previous framework. According to the present definitions, a semiexact category is a category equipped with an ideal of ‘null’ morphisms and provided with kernels and cokernels with respect to this ideal. A homological category satisfies some further conditions that allow the construction of subquotients and induced morphisms, in particular the homology of a chain complex or the spectral sequence of an exact couple. Extending abelian categories, and also the p-exact ones, these notions include the usual domains of homology and homotopy theories, e.g. the category of ‘pairs’ of topological spaces or groups; they also include their codomains, since the sequences of homotopy ‘objects’ for a pair of pointed spaces or a fibration can be viewed as exact sequences in a homological category, whose objects are actions of groups on pointed sets. Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups Contents:IntroductionSemiexact categoriesHomological CategoriesSubquotients, Homology and Exact CouplesSatellitesUniversal ConstructionsApplications to Algebraic TopologyHomological Theories and Biuniversal ModelsAppendix A. Some Points of Category Theory Readership: Graduate students, professors and researchers in pure mathematics, in particular category theory and algebraic topology. Keywords:Non Abelian Homological Algebra;Spectral Sequences;Distributive Lattices;Orthodox Semigroups;Categories of RelationsReviews: “The range of applications and examples is considerable and many are outside the reach of more standard forms of homological algebra, but the methods used here also give insight as to 'why' the classical theory works and how its results can be interpreted.” Zentralblatt MATH