Hopf Algebras And Their Generalizations From A Category Theoretical Point Of View

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Hopf Algebras and Their Generalizations from a Category Theoretical Point of View

Author : Gabriella Böhm
Publisher : Springer
ISBN 13 : 3319981374
Total Pages : 165 pages
Book Rating : 4.3/5 (199 download)

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Book Synopsis Hopf Algebras and Their Generalizations from a Category Theoretical Point of View by : Gabriella Böhm

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.

Hopf Algebras and Generalizations

Author : Louis H. Kauffman
Publisher : American Mathematical Soc.
ISBN 13 : 0821838202
Total Pages : 186 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Hopf Algebras and Generalizations by : Louis H. Kauffman

Download or read book Hopf Algebras and Generalizations written by Louis H. Kauffman and published by American Mathematical Soc.. This book was released on 2007 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hopf algebras have proved to be very interesting structures with deep connections to various areas of mathematics, particularly through quantum groups. Indeed, the study of Hopf algebras, their representations, their generalizations, and the categories related to all these objects has an interdisciplinary nature. It finds methods, relationships, motivations and applications throughout algebra, category theory, topology, geometry, quantum field theory, quantum gravity, and also combinatorics, logic, and theoretical computer science. This volume portrays the vitality of contemporary research in Hopf algebras. Altogether, the articles in the volume explore essential aspects of Hopf algebras and some of their best-known generalizations by means of a variety of approaches and perspectives. They make use of quite different techniques that are already consolidated in the area of quantum algebra. This volume demonstrates the diversity and richness of its subject. Most of its papers introduce the reader to their respective contexts and structures through very expository preliminary sections.

Bimonoids for Hyperplane Arrangements

Author : Marcelo Aguiar
Publisher : Cambridge University Press
ISBN 13 : 1108852785
Total Pages : 854 pages
Book Rating : 4.1/5 (88 download)

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Book Synopsis Bimonoids for Hyperplane Arrangements by : Marcelo Aguiar

Download or read book Bimonoids for Hyperplane Arrangements written by Marcelo Aguiar and published by Cambridge University Press. This book was released on 2020-03-19 with total page 854 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this monograph is to develop Hopf theory in a new setting which features centrally a real hyperplane arrangement. The new theory is parallel to the classical theory of connected Hopf algebras, and relates to it when specialized to the braid arrangement. Joyal's theory of combinatorial species, ideas from Tits' theory of buildings, and Rota's work on incidence algebras inspire and find a common expression in this theory. The authors introduce notions of monoid, comonoid, bimonoid, and Lie monoid relative to a fixed hyperplane arrangement. They also construct universal bimonoids by using generalizations of the classical notions of shuffle and quasishuffle, and establish the Borel–Hopf, Poincaré–Birkhoff–Witt, and Cartier–Milnor–Moore theorems in this setting. This monograph opens a vast new area of research. It will be of interest to students and researchers working in the areas of hyperplane arrangements, semigroup theory, Hopf algebras, algebraic Lie theory, operads, and category theory.

Hopf Algebras, Quantum Groups and Yang-Baxter Equations

Author : Florin Felix Nichita
Publisher : MDPI
ISBN 13 : 3038973246
Total Pages : 239 pages
Book Rating : 4.0/5 (389 download)

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Book Synopsis Hopf Algebras, Quantum Groups and Yang-Baxter Equations by : Florin Felix Nichita

Download or read book Hopf Algebras, Quantum Groups and Yang-Baxter Equations written by Florin Felix Nichita and published by MDPI. This book was released on 2019-01-31 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a printed edition of the Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" that was published in Axioms

Classical Hopf Algebras and Their Applications

Author : Pierre Cartier
Publisher : Springer Nature
ISBN 13 : 3030778452
Total Pages : 277 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis Classical Hopf Algebras and Their Applications by : Pierre Cartier

Download or read book Classical Hopf Algebras and Their Applications written by Pierre Cartier and published by Springer Nature. This book was released on 2021-09-20 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is dedicated to the structure and combinatorics of classical Hopf algebras. Its main focus is on commutative and cocommutative Hopf algebras, such as algebras of representative functions on groups and enveloping algebras of Lie algebras, as explored in the works of Borel, Cartier, Hopf and others in the 1940s and 50s. The modern and systematic treatment uses the approach of natural operations, illuminating the structure of Hopf algebras by means of their endomorphisms and their combinatorics. Emphasizing notions such as pseudo-coproducts, characteristic endomorphisms, descent algebras and Lie idempotents, the text also covers the important case of enveloping algebras of pre-Lie algebras. A wide range of applications are surveyed, highlighting the main ideas and fundamental results. Suitable as a textbook for masters or doctoral level programs, this book will be of interest to algebraists and anyone working in one of the fields of application of Hopf algebras.

Hopf Algebras and Root Systems

Author : István Heckenberger
Publisher : American Mathematical Soc.
ISBN 13 : 1470452324
Total Pages : 582 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Hopf Algebras and Root Systems by : István Heckenberger

Download or read book Hopf Algebras and Root Systems written by István Heckenberger and published by American Mathematical Soc.. This book was released on 2020-06-19 with total page 582 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to Hopf algebras in braided monoidal categories with applications to Hopf algebras in the usual sense. The main goal of the book is to present from scratch and with complete proofs the theory of Nichols algebras (or quantum symmetric algebras) and the surprising relationship between Nichols algebras and generalized root systems. In general, Nichols algebras are not classified by Cartan graphs and their root systems. However, extending partial results in the literature, the authors were able to associate a Cartan graph to a large class of Nichols algebras. This allows them to determine the structure of right coideal subalgebras of Nichols systems which generalize Nichols algebras. As applications of these results, the book contains a classification of right coideal subalgebras of quantum groups and of the small quantum groups, and a proof of the existence of PBW-bases that does not involve case by case considerations. The authors also include short chapter summaries at the beginning of each chapter and historical notes at the end of each chapter. The theory of Cartan graphs, Weyl groupoids, and generalized root systems appears here for the first time in a book form. Hence, the book serves as an introduction to the modern classification theory of pointed Hopf algebras for advanced graduate students and researchers working in categorial aspects and classification theory of Hopf algebras and their generalization.

Generalized Lie Theory in Mathematics, Physics and Beyond

Author : Sergei D. Silvestrov
Publisher : Springer Science & Business Media
ISBN 13 : 3540853324
Total Pages : 308 pages
Book Rating : 4.5/5 (48 download)

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Book Synopsis Generalized Lie Theory in Mathematics, Physics and Beyond by : Sergei D. Silvestrov

Download or read book Generalized Lie Theory in Mathematics, Physics and Beyond written by Sergei D. Silvestrov and published by Springer Science & Business Media. This book was released on 2008-11-18 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the cutting edge of the fundamental role of generalizations of Lie theory and related non-commutative and non-associative structures in mathematics and physics.

Algebraic Groups and Their Generalizations: Classical Methods

Author : William Joseph Haboush
Publisher : American Mathematical Soc.
ISBN 13 : 0821815407
Total Pages : 397 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Algebraic Groups and Their Generalizations: Classical Methods by : William Joseph Haboush

Download or read book Algebraic Groups and Their Generalizations: Classical Methods written by William Joseph Haboush and published by American Mathematical Soc.. This book was released on 1994 with total page 397 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations

Author : Stefaan Caenepeel
Publisher : Springer
ISBN 13 : 3540480420
Total Pages : 354 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations by : Stefaan Caenepeel

Download or read book Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations written by Stefaan Caenepeel and published by Springer. This book was released on 2004-10-14 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: Doi-Koppinen Hopf modules and entwined modules unify various kinds of modules that have been intensively studied over the past decades, such as Hopf modules, graded modules, Yetter-Drinfeld modules. The book presents a unified theory, with focus on categorical concepts generalizing the notions of separable and Frobenius algebras, and discussing relations with smash products, Galois theory and descent theory. Each chapter of Part II is devoted to a particular nonlinear equation. The exposé is organized in such a way that the analogies between the four are clear: the quantum Yang-Baxter equation is related to Yetter-Drinfeld modules, the pentagon equation to Hopf modules, and the Long equation to Long dimodules. The Frobenius-separability equation provides a new viewpoint to Frobenius and separable algebras.

Quasi-Hopf Algebras

Author : Daniel Bulacu
Publisher : Cambridge University Press
ISBN 13 : 1108427014
Total Pages : 545 pages
Book Rating : 4.1/5 (84 download)

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Book Synopsis Quasi-Hopf Algebras by : Daniel Bulacu

Download or read book Quasi-Hopf Algebras written by Daniel Bulacu and published by Cambridge University Press. This book was released on 2019-02-21 with total page 545 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained book dedicated to Drinfeld's quasi-Hopf algebras takes the reader from the basics to the state of the art.

Galois Theory, Hopf Algebras, and Semiabelian Categories

Author : George Janelidze
Publisher : American Mathematical Soc.
ISBN 13 : 0821832905
Total Pages : 582 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Galois Theory, Hopf Algebras, and Semiabelian Categories by : George Janelidze

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.

Rings, Hopf Algebras, and Brauer Groups

Author : Stefaan Caenepeel
Publisher : CRC Press
ISBN 13 : 1000153282
Total Pages : 352 pages
Book Rating : 4.0/5 (1 download)

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Book Synopsis Rings, Hopf Algebras, and Brauer Groups by : Stefaan Caenepeel

Download or read book Rings, Hopf Algebras, and Brauer Groups written by Stefaan Caenepeel and published by CRC Press. This book was released on 2020-09-29 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Based on papers presented at a recent international conference on algebra and algebraic geometry held jointly in Antwerp and Brussels, Belgium. Presents both survey and research articles featuring new results from the intersection of algebra and geometry. "

Galois Theory, Hopf Algebras, and Semiabelian Categories

Author : George Janelidze, Bodo Pareigis, and Walter Tholen
Publisher : American Mathematical Soc.
ISBN 13 : 9780821871478
Total Pages : 588 pages
Book Rating : 4.8/5 (714 download)

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Book Synopsis Galois Theory, Hopf Algebras, and Semiabelian Categories by : George Janelidze, Bodo Pareigis, and Walter Tholen

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:

Progress in Physics, vol. 1/2005

Author : Dmitri Rabounski
Publisher : Infinite Study
ISBN 13 :
Total Pages : 88 pages
Book Rating : 4./5 ( download)

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Book Synopsis Progress in Physics, vol. 1/2005 by : Dmitri Rabounski

Download or read book Progress in Physics, vol. 1/2005 written by Dmitri Rabounski and published by Infinite Study. This book was released on with total page 88 pages. Available in PDF, EPUB and Kindle. Book excerpt: Progress in Physics has been created for publications on advanced studies in theoretical and experimental physics, including related themes from mathematics.

Algebraic Geometry for Associative Algebras

Author : Freddy Van Oystaeyen
Publisher : CRC Press
ISBN 13 : 1482270528
Total Pages : 302 pages
Book Rating : 4.4/5 (822 download)

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Book Synopsis Algebraic Geometry for Associative Algebras by : Freddy Van Oystaeyen

Download or read book Algebraic Geometry for Associative Algebras written by Freddy Van Oystaeyen and published by CRC Press. This book was released on 2000-06-06 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work focuses on the association of methods from topology, category and sheaf theory, algebraic geometry, noncommutative and homological algebras, quantum groups and spaces, rings of differential operation, Cech and sheaf cohomology theories, and dimension theories to create a blend of noncommutative algebraic geometry. It offers a scheme theor

Monoidal Functors, Species and Hopf Algebras

Author : Marcelo Aguiar
Publisher : American Mathematical Soc.
ISBN 13 : 9780821847763
Total Pages : 784 pages
Book Rating : 4.8/5 (477 download)

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Book Synopsis Monoidal Functors, Species and Hopf Algebras by : Marcelo Aguiar

Download or read book Monoidal Functors, Species and Hopf Algebras written by Marcelo Aguiar and published by American Mathematical Soc.. This book was released on 2010 with total page 784 pages. Available in PDF, EPUB and Kindle. Book excerpt: This research monograph integrates ideas from category theory, algebra and combinatorics. It is organized in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students.

Non-commutative and Non-associative Algebra and Analysis Structures

Author : Sergei Silvestrov
Publisher : Springer Nature
ISBN 13 : 3031320093
Total Pages : 833 pages
Book Rating : 4.0/5 (313 download)

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Book Synopsis Non-commutative and Non-associative Algebra and Analysis Structures by : Sergei Silvestrov

Download or read book Non-commutative and Non-associative Algebra and Analysis Structures written by Sergei Silvestrov and published by Springer Nature. This book was released on 2023-09-25 with total page 833 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of the 2019 conference on Stochastic Processes and Algebraic Structures held in SPAS2019, Västerås, Sweden, from September 30th to October 2nd 2019 was to showcase the frontiers of research in several important topics of mathematics, mathematical statistics, and its applications. The conference has been organized along the following tracks: 1. Stochastic processes and modern statistical methods in theory and practice, 2. Engineering Mathematics, 3. Algebraic Structures and applications. This book highlights the latest advances in algebraic structures and applications focused on mathematical notions, methods, structures, concepts, problems, algorithms, and computational methods for the natural sciences, engineering, and modern technology. In particular, the book features mathematical methods and models from non-commutative and non-associative algebras and rings associated to generalizations of differential calculus, quantum deformations of algebras, Lie algebras, Lie superalgebras, color Lie algebras, Hom-algebras and their n-ary generalizations, semi-groups and group algebras, non-commutative and non-associative algebras and computational algebra interplay with q-special functions and q-analysis, topology, dynamical systems, representation theory, operator theory and functional analysis, applications of algebraic structures in coding theory, information analysis, geometry and probability theory. The book gathers selected, high-quality contributed chapters from several large research communities working on modern algebraic structures and their applications. The chapters cover both theory and applications, and are illustrated with a wealth of ideas, theorems, notions, proofs, examples, open problems, and results on the interplay of algebraic structures with other parts of Mathematics. The applications help readers grasp the material, and encourage them to develop new mathematical methods and concepts in their future research. Presenting new methods and results, reviews of cutting-edge research, open problems, and directions for future research, will serve as a source of inspiration for a broad range of researchers and students.