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.

Author: Robert G. Underwood

Publisher: Springer

ISBN: 9783319189918

Category: Mathematics

Page: 150

View: 597

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.

This book offers a unified description of Hopf algebras and their generalizations from a category theoretical point of view.

Author: Gabriella Böhm

Publisher: Springer

ISBN: 9783319981376

Category: Mathematics

Page: 165

View: 280

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.

Extending Structures: Fundamentals and Applications treats the extending structures (ES) problem in the context of groups, Lie/Leibniz algebras, associative algebras and Poisson/Jacobi algebras.

Author: Ana Agore

Publisher: CRC Press

ISBN: 9781351168717

Category: Mathematics

Page: 224

View: 433

Extending Structures: Fundamentals and Applications treats the extending structures (ES) problem in the context of groups, Lie/Leibniz algebras, associative algebras and Poisson/Jacobi algebras. This concisely written monograph offers the reader an incursion into the extending structures problem which provides a common ground for studying both the extension problem and the factorization problem. Features Provides a unified approach to the extension problem and the factorization problem Introduces the classifying complements problem as a sort of converse of the factorization problem; and in the case of groups it leads to a theoretical formula for computing the number of types of isomorphisms of all groups of finite order that arise from a minimal set of data Describes a way of classifying a certain class of finite Lie/Leibniz/Poisson/Jacobi/associative algebras etc. using flag structures Introduces new (non)abelian cohomological objects for all of the aforementioned categories As an application to the approach used for dealing with the classification part of the ES problem, the Galois groups associated with extensions of Lie algebras and associative algebras are described

... coquasitriangular bialgebra on a coquasitriangular coalgebra, 462 free Hopf algebra on a coalgebra, 230 free pointed ... 157 free pointed irreducible Hopf algebra on a Yetter Drinfel'd module, 487 functor, 35 contravariant, 51 fundamental ...

Author: David E. Radford

Publisher: World Scientific

ISBN: 9789814335997

Category: Mathematics

Page: 559

View: 577

The book provides a detailed account of basic coalgebra and Hopf algebra theory with emphasis on Hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups. It is intended to be a graduate text as well as a research monograph.

that there are exactly p + 1 non-isomorphic semisimple Hopf algebras of
dimension p” which are neither group algebras nor duals of group algebras. A
great boost to the theory of (finite-dimensional) Hopf algebras was given by
Drinfeld in the ...

Author: Karen Redrobe Beckman

Publisher: American Mathematical Soc.

ISBN: 9780821820360

Category: Mathematics

Page: 569

View: 766

The arithmetic and geometry of moduli spaces and their fundamental groups are a very active research area. This book offers a complete overview of developments made over the last decade. The papers in this volume examine the geometry of moduli spaces of curves with a function on them. The main players in Part 1 are the absolute Galois group $G_{\mathbb Q}$ of the algebraic numbers and its close relatives. By analyzing how $G_{\mathbb Q}$ acts on fundamental groups defined by Hurwitz moduli problems, the authors achieve a grand generalization of Serre's program from the 1960s.Papers in Part 2 apply $\theta$-functions and configuration spaces to the study of fundamental groups over positive characteristic fields. In this section, several authors use Grothendieck's famous lifting results to give extensions to wildly ramified covers. Properties of the fundamental groups have brought collaborations between geometers and group theorists. Several Part 3 papers investigate new versions of the genus 0 problem. In particular, this includes results severely limiting possible monodromy groups of sphere covers. Finally, Part 4 papers treat Deligne's theory of Tannakian categories and arithmetic versions of the Kodaira-Spencer map. This volume is geared toward graduate students and research mathematicians interested in arithmetic algebraic geometry.

This is the first book to be dedicated entirely to Drinfeld's quasi-Hopf algebras.

Author: Daniel Bulacu

Publisher: Cambridge University Press

ISBN: 9781108632652

Category: Mathematics

Page:

View: 613

This is the first book to be dedicated entirely to Drinfeld's quasi-Hopf algebras. Ideal for graduate students and researchers in mathematics and mathematical physics, this treatment is largely self-contained, taking the reader from the basics, with complete proofs, to much more advanced topics, with almost complete proofs. Many of the proofs are based on general categorical results; the same approach can then be used in the study of other Hopf-type algebras, for example Turaev or Zunino Hopf algebras, Hom-Hopf algebras, Hopfish algebras, and in general any algebra for which the category of representations is monoidal. Newcomers to the subject will appreciate the detailed introduction to (braided) monoidal categories, (co)algebras and the other tools they will need in this area. More advanced readers will benefit from having recent research gathered in one place, with open questions to inspire their own research.

This book provides an introduction to the theory of quantum groups with emphasis on their duality and on the setting of operator algebras. The book is addressed to graduate students and non-experts from other fields.

Author: Thomas Timmermann

Publisher: European Mathematical Society

ISBN: 3037190434

Category: Mathematics

Page: 407

View: 904

This book provides an introduction to the theory of quantum groups with emphasis on their duality and on the setting of operator algebras. The book is addressed to graduate students and non-experts from other fields. Only basic knowledge of (multi-)linear algebra is required for the first part, while the second and third part assume some familiarity with Hilbert spaces, CÝsuperscript *¨-algebras, and von Neumann algebras.

This book is addressed to graduate students and research workers in theoretical physics who want a thorough introduction to group theory and Hopf algebras.

Author: George Janelidze, Bodo Pareigis, and Walter TholenPublish On:

The goals of this paper are (1) to compare the Galois groupoid that appears
naturally in the construction of the coverings fundamental groupoid topos given
by Bunge (1992) with the formal Galois groupoid defined by Janelidze (1990) in
a very ...

Author: George Janelidze, Bodo Pareigis, and Walter Tholen

Publisher: American Mathematical Soc.

ISBN: 0821871471

Category: Mathematics

Page:

View: 190

Survey and research papers in this volume are based on talks given at a workshop held at The Fields Institute for Research in the Mathematical Sciences (Toronto, ON, Canada). It provides an up-to-date account by leading researchers on the many current connections among Galois theories, Hopf algebras, and semiabelian categories.

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions.

Author: Christian Kassel

Publisher: Springer Science & Business Media

ISBN: 9781461207832

Category: Mathematics

Page: 534

View: 115

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.

This is the first-ever textbook on the Yang-Baxter equation.

Author: Zhongqi Ma

Publisher: World Scientific

ISBN: 9810213832

Category: Science

Page: 318

View: 534

This is the first-ever textbook on the Yang-Baxter equation. A key nonlinear equation for solving two important models in many-body statistical theory - the many-body problem in one dimension with repulsive delta-function interaction presented by Professor Baxter in 1972 - it has become one of the main concerns of physicists and mathematicians in the last ten years. A textbook on this subject which also serves as a reference book is vital for an equation which plays important roles in diverse areas of physics and mathematics like the completely integrable statistical models, conformal field theories, topological field theories, the theory of braid groups, the theory of knots and links, etc. This book arose from lectures given by the author in an attempt to reformulate the results of the rapidly developing research and make the material more accessible. It explains the presentation of the Yang-Baxter equation from statistical models, and expound systematically the meaning and methods of solving for this equation. From the viewpoint of theoretical physics it aims to develop an intuitive understanding of the fundamental knowledge of the Hopf algebras, quantization of Lie bialgebras, and the quantum enveloping algebras, and places emphasis on the introduction of the calculation skill in terms of the physical language.

This is the first of two volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras.

Author: Andrzej Skowroński

Publisher: European Mathematical Society

ISBN: 3037191023

Category: Mathematics

Page: 650

View: 906

This is the first of two volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book serves as a general introduction to basic results and techniques of the modern representation theory of finite dimensional associative algebras over fields, including the Morita theory of equivalences and dualities and the Auslander-Reiten theory of irreducible morphisms and almost split sequences. The second part is devoted to fundamental classical and recent results concerning the Frobenius algebras and their module categories. Moreover, the prominent classes of Frobenius algebras, the Hecke algebras of Coxeter groups, and the finite dimensional Hopf algebras over fields are exhibited. This volume is self contained and the only prerequisite is a basic knowledge of linear algebra. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well as mathematicians working in other fields.

This book, addressing scientists and postgraduates, contains a detailed and rather complete presentation of the algebraic framework.

Author: Ludwig Pittner

Publisher: Springer Science & Business Media

ISBN: 9783540478010

Category: Science

Page: 469

View: 590

Quantum groups and quantum algebras as well as non-commutative differential geometry are important in mathematics and considered to be useful tools for model building in statistical and quantum physics. This book, addressing scientists and postgraduates, contains a detailed and rather complete presentation of the algebraic framework. Introductory chapters deal with background material such as Lie and Hopf superalgebras, Lie super-bialgebras, or formal power series. Great care was taken to present a reliable collection of formulae and to unify the notation, making this volume a useful work of reference for mathematicians and mathematical physicists.

Elementary properties of Hopf algebras . 5 . ... zero the category of graded Lie
algebras is isomorphic with the category of primitively generated Hopf algebras .
... It is here that one finds the fundamental theorem of Hopf , Leray , and Borel .

A great deal of emphasis is placed on the coalgebra which is the dual of n x n matrices over a field. This is the most basic example of a coalgebra for our purposes and is at the heart of most algebraic constructions described in this book.

Author: L.A. Lambe

Publisher: Springer Science & Business Media

ISBN: 9781461541097

Category: Mathematics

Page: 300

View: 305

Chapter 1 The algebraic prerequisites for the book are covered here and in the appendix. This chapter should be used as reference material and should be consulted as needed. A systematic treatment of algebras, coalgebras, bialgebras, Hopf algebras, and represen tations of these objects to the extent needed for the book is given. The material here not specifically cited can be found for the most part in [Sweedler, 1969] in one form or another, with a few exceptions. A great deal of emphasis is placed on the coalgebra which is the dual of n x n matrices over a field. This is the most basic example of a coalgebra for our purposes and is at the heart of most algebraic constructions described in this book. We have found pointed bialgebras useful in connection with solving the quantum Yang-Baxter equation. For this reason we develop their theory in some detail. The class of examples described in Chapter 6 in connection with the quantum double consists of pointed Hopf algebras. We note the quantized enveloping algebras described Hopf algebras. Thus for many reasons pointed bialgebras are elsewhere are pointed of fundamental interest in the study of the quantum Yang-Baxter equation and objects quantum groups.

Contents include: Hochschild Homology of Function Algebras Associated with Singularities; On the KK-Theory of Stable Projective Limits; Noncommutative Integrability; Gauge Invariance of the Chern-Simons Action in Noncommutative Geometry; ...

Author: Daniel Kastler

Publisher: Nova Science Pub Incorporated

ISBN: STANFORD:36105023609014

Category: Science

Page: 461

View: 712

Contents include: Hochschild Homology of Function Algebras Associated with Singularities; On the KK-Theory of Stable Projective Limits; Noncommutative Integrability; Gauge Invariance of the Chern-Simons Action in Noncommutative Geometry; The Analysis of the Hochshild Homology; Coproducts and Operations on Cyclic Cohomology; Powers of Quantum Matrices and Relations Between Them; Introductory Notes on Extensions of Hopf Algebras; Hopf Algebras from the Quantum Geometry Point of View; Equation Pentagonale, Bige bres et Espaces de Modules; Chiral Anomalies in the Spectral Action; Standard Model and Unimodularity Condition; On Feynman Graphs as Elements of a Hopf Algebra.

8 . N . BOURBAKI , Algebra , Hermann , Paris , 1970 , Chapter III . 9 . H . CRAPO
and G . - C . ROTA , Combinatorial Geometries , M . I . T . Press , Cambridge ,
Mass . , 1971 . 10 . J . A . DIEUDONNÉ , Introduction to the Theory of Formal
Groups ...

A - module whose extension to C is just N . SECTION 2 . 2 FUNDAMENTAL
THEOREM ON COALGEBRAS We conclude this chapter with an important
application of rational modules . Suppose ( El consists of subcoalgebras of a
coalgebra .