Hopf Algebras (Cambridge Tracts in Mathematics)

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The concept of Hopf algebras was first introduced in the theory of algebraic topology but in recent years has been developed by many mathematicians and applied to other areas of mathematics such as Lie groups, algebraic groups and Galois theory. This book is an introduction to the basic theory of Hopf algebras for the reader already familiar with the basic ideas of linear algebra and commutative algebra. After introducing and discussing the basic properties of coalgebras, bialgebras and Hopf algebras, the author treats the fundamental structure theorem of bi-modules and Sullivan’s proof of the existence and uniqueness of integrals of Hopf algebras. This book will interest graduate students and research workers who specialise in algebra.

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Intermediate Algebra
Ignacio Bello , and Fran Hopf
Manufacturer: McGraw-Hill Science/Engineering/Math
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Binding: Paperback

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ASIN: 0072991003

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Intermediate Algebra prepares students for further courses in the college math curriculum. Students of all backgrounds will be delighted to find a refreshing book that appeals to every learning style and reaches out to diverse demographics. Through down-to-earth explanations, patient skill-building, and exceptionally interesting and realistic applications, this worktext will empower students to learn and master algebra in the real world.

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SSM f/u w/ Bello, Intermediate Algebra
Ignacio Bello , and Fran Hopf
Manufacturer: McGraw-Hill Science/Engineering/Math
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Intermediate Algebra

ASIN: 0072945583

Coxeter Groups and Hopf Algebras (Fields Institute Monographs,) (Fields Institute Monographs,)

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Coxeter Groups and Hopf Algebras (Fields Institute Monographs,) (Fields Institute Monographs,)
Marcelo Aguiar
Manufacturer: American Mathematical Society
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Binding: Hardcover

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An important idea in the work of G.-C. Rota is that certain combinatorial objects give rise to Hopf algebras that reflect the manner in which these objects compose and decompose. Recent work has seen the emergence of several interesting Hopf algebras of this kind, which connect diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This monograph presents a novel geometric approach using Coxeter complexes and the projection maps of Tits for constructing and studying many of these objects as well as new ones. The first three chapters introduce the necessary background ideas making this work accessible to advanced graduate students. The later chapters culminate in a unified and conceptual construction of several Hopf algebras based on combinatorial objects which emerge naturally from the geometric viewpoint. This work lays a foundation and provides new insights for further development of the subject. Titles in this series are copublished with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Quantum Groups (Graduate Texts in Mathematics)

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For quantum mathematicians
Kassel's Quantum Groups

Quantum Groups (Graduate Texts in Mathematics)
Christian Kassel
Manufacturer: Springer
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ASIN: 0387943706

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This book provides an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and on Drinfeld's recent fundamental contributions. The first part presents in detail the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Part Two focuses on Hopf algebras that produce solutions of the Yang-Baxter equation, and on Drinfeld's quantum double construction. In the following part we construct isotopy invariants of knots and links in the three-dimensional Euclidean space, using the language of tensor categories. The last part is an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations, culminating in the construction of Kontsevich's universal knot invariant.

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For quantum mathematicians.......2001-09-03

This book is further evidence of the tremendous influence that quantum physics, especially quantum field theory and superstring theory, has had on modern mathematics. Very rich mathematical structures and simplified methods of proof have resulted from looking at mathematics from a quantum point of view. Because of the enormous success of viewpoint, examples being proofs of the Atiyah-Singer index theorem, the Jones polynomial, and the Seiberg-Witten equations, one could justify a rephrasing of the remark by Eugene Wigner and now speak of "the reasonable effectiveness of physics in mathematics".
The book gives a fine overview of a field that has only been around for a few decades, and is manifested by brilliant developments. Those who have worked with the Yang-Baxter equations from the theory of exactly solved models in statistical mechanics will see these equations come alive here in a much more general form. In addition, knot theorists and geometric topologists will appreciate the discussion of how their constructions can be cast in the tensor and tangle categories that are explained in detail in this book. The title of the book is a little strange, given that the structures treated are more specific than groups, but the author has explained well the theory of quantum groups, as is it is now referrred to in journal classification schemes.

An in-depth reading of the book is time-consuming, and no doubt the average reader will not read it from cover to cover but instead will peruse only the areas of immediate interest. Part One of the book is an overview of what the author calls quantum SL(2), which is an example of a Hopf algebra. The first two chapters are purely a review of algebra, with the third being an introduction to coalgebras, which the author, in a categorical sense, identifies as being dual to an algebra. The notion of a bialgebra is also discussed, which is essentially a vector space equipped with both an algebra structure and a coalgebra structure. Taking a tensor product of this vector space with itself and examining certain morphisms between these structures gives a set of compatibility conditions that define the bialgebra structure. A Hopf algebra is then a bialgebra that has a special endomorphism of the underlying vector space. The algebraic topologist reader will be familiar with Hopf algebras via studies of product manifolds such as Lie groups. Quantum groups have given many examples of non-commutative non-cocommutative bialgebras than were known before this research area had taken off. The author also discusses the quantum plane as an object that generalizes the affine plane, namely the two variables x, y generating the plane no longer commute but instead satisfy yx = q xy. The author investigates in detail the quantum group SLq(n), which is based on the classical Lie group. References are given for quantum groups based on the other Lie groups, such as the orthogonal and symplectic groups. The Lie algebra Uq(sl(2)) is given a detailed treatment by the author when q is not a root of unity. This Hopf algebra is a 1-parameter deformation of the enveloping algebra of the Lie algebra sl(2) considered in earlier chapters. The reader interested in the renormalization is strongly urged to read this first part, as recently it has been shown that for any quantum field theory, the combinatorics of Feynman diagrams gives rise to a Hopf algebra which is commutative as an algebra, and is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by one particle irreducible Feynman diagrams. The Lie bracket of two diagrams is computed from insertions of one graph in the other and vice versa, and the Lie group G is the group of characters of the Hopf algebra. This structure is used to go on and formulate the renormalization problem rigorously.

Part two is an overview of the famous Yang-Baxter equation whose exact solutions in terms of R-matrices have generated a vast amount of research. The author introduces the concept of a braided bialgebra, which contain a "universal" R-matrix which induces a solution of the Yang-Baxter equation on all of their modules, and thus giving a systematic method for constructing solutions of the Yang-Baxter equation. The duals of these bialgebras give a cobraided bialgebra, and the author shows how to construct a cobraided bialgebra out of any solution of the Yang-Baxter equation. It is also shown how the quantum groups GLq(2) and SLq(2) can be obtained by this method, and it is proven that they are cobraided. The famous Drinfeld quantum double, yielding a braided Hopf algebra out of any finite-dimensional Hopf algebra with invertible antipode, is discussed in great detail.

The next part is basically low-dimensional topology in the form of knots, links, and braids, wherein the author discusses the relationship between the Jones polynomial and R-matrices. The connection between knot theory and quantum groups is given by the representation theory of Hopf algebras, this connection taking place in the tensor category. A certain strict tensor category is built out of tangles, and shown to give isotopy invariants of links. Braiding in the tensor category is used to formalize the notion of crossing in link and tangle diagrams. Tensor categories modeled on framed tangles or "ribbons" are introduced to illustrate duality. The concept of a quasi-bialgebra is introduced and braid group representations of these are constructed. When quasi-bialgebras are equivalent under a "gauge transformation" introduced here, they have the same braid group representation.

The last part considers the role of monodromy in the theory of quantum groups. The quantum enveloping algebras due to Drinfeld and Jimbo are discussed and shown to provide isotopy invariants of links. The monodromy of the Knizhnik-Zamolodchikov system is shown to be equivalent to the braid group representation of this system. Knot invariants of finite type are shown to be universal invariants for quantum groups.

Kassel's Quantum Groups.......2000-09-13

This is a very well-written book covering not only quantum groups but also the connections with low-dimensional topology. When comparing the presentation in the book with other presentations in the literature I realized how much work the author put in trying to make the material accessible. Part I covers Hopf algebras and quantum SL(2). Part II covers the Yang-Baxter equation and its solutions (R-matrices) and Drinfeld's quantum double construction. Part III deals with the applications to low-dimensional topology, i.e., the construction of knot invariants via braided tensor categories, in particular the construction of the Jones-Conway polynomial. Part IV is an account of Drinfeld's treatment of the monodromy of the Knizhnik-Zamolodchikov equations. The last chapter discusses finite type knot invariants, and discusses Cartier's combinatorial version of Kontsevich's universal knot invariant.

Rational Homotopical Models and Uniqueness (Memoirs of the American Mathematical Society)

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Rational Homotopical Models and Uniqueness (Memoirs of the American Mathematical Society)
Martin Majewski
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Paperback

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Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory (Mathematical Surveys and Monographs)

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Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory (Mathematical Surveys and Monographs)
Lindsay Childs
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Hardcover

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ASIN: 0821821318

Book Description

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 $p^2$; and Kummer theory of formal groups.

Book Description

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.

Advances in Hopf Algebras (Lecture Notes in Pure and Applied Mathematics)

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Advances in Hopf Algebras (Lecture Notes in Pure and Applied Mathematics)

Manufacturer: CRC
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Binding: Paperback

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"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. "

Brauer Groups, Hopf Algebras and Galois Theory (K-Monographs in Mathematics)

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Brauer Groups, Hopf Algebras and Galois Theory (K-Monographs in Mathematics)
S. Caenepeel
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 079234829X

Book Description

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.

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In this almost documentary account of his own experiences of penal servitude in Serbia, Dostoevsky describes the physical and mental suffering of the convicts, the squalor and the degradation, in relentless detail. The inticate procedure whereby the men strip for the bath without removing their ten-pound leg-fetters is an extraordinary tour de force, compared by Turgenev to passages from Dante's Inferno. Terror and resignation - the rampages of a pyschopath, the brief serence interlude of Christmas Day - are evoked by Dostoevsky, writing several years after his release, with a strikingly uncharacteristic detachment. For this reason, House of the Dead is certainly the least Dostoevskian of his works, yet, paradoxically, it ranks among his great masterpieces.

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a note on the translation.......2007-08-10

I wanted to say that the edition published by Oxford Press called "Memoirs from the House of the Dead" is translated by Jessie Coulson. I have no idea who Coulson was but he is an underrated Dostoevsky translator. I have read this and his "Crime and Punishment" (which is the Norton Critical edition) and I think he was much better than most of the transltors like Garnett, Ginzburg, McDuff and the rest. There are many traslations of this book available and I think this one gets left off bookstore shelves without reason. It is a very passionate translation.

Surviving the House of the Dead.......2007-06-20

The "House of the Dead" is an early semi autobiographical work of Fyodor Dostoevsky, telling the tale of a nobleman who is imprisoned in a labour camp in Siberia for a crime of passion. The tale is semi-autobiographical because Dostoevsky as a young man was also imprisoned in Siberia for being a member of a radical political organisation an experience which was to form and influence his amazing insights and understanding of human nature.

Although not Dostoevsky greatest work "House of the Dead" is still a fascinating portrait of life in the Tsarist gulags system - a life of great hardship and deprivation yet filled with simple moments of humanity showing mankinds ability to adapt and survive in the most extreme of circumstances. Dostoevsky tells his story in a chronological order from his characters arrival as a new alienated and withdrawn noble to his gradual adjustment to prison and the return of hope as he realises that he can survive and will have a life after the completion of his term.

It is also interesting to read House of the Dead in conjunction with later works such as One Day in the Life of Ivan Denisovich by Aleksandr Solzhenitsyn dealing with the gulag system in communist times to see the continuation of the institution despite the changing of social regimes.

From Siberia with Love: A memoir of a frozen prison hell by the great psychological novelist Fyodor Dostoyevsky.......2007-05-10

Fyodor Dostoyevsky (1821-1881) wrote "From the House of the Dead" in 1860. It is a fictionalized account of the four years (1850-54) he spent in a penal colony in distant Siberia. Dostoyevsky had been sentenced due to his involvement in a plot to assasinate the Tsar. Following his imprisonment he served in the Russian army; returned to European Russia writing such classics as "The Brothers Karamazov" and "Crime and Punishment."
"From the House of the Dead" is a lesser known but still classic account of the torment of the prisoner's life in a totalitarian state.
We learn the horror of labor in the subzero work camp; the stories of several of the prisoners; animals who lived in the area and the freezing isolation and pain of countless days of misery. Dostoyevsky was a young intellectual forced to live, eat and sleep with men who came from a peasant background of cruelty, coarseness and brutality. Many of the camp's officials were sadistic and cruel in their treatment of the wretches whose lives they ruled with an iron fist.
Dostoyevsky is able to look through the keyhole of the human soul in all its multifaceted complexity. His descriptions of the bleak landscape is journalistic in its detail.
I have always loved Dostoyevsky's major novels. This was a new one for me and I am glad I read it. I consider it imperative perusing for anyone who wants to know what kind of man Dostoevsky was. In the last lines of the book he reports the main character's release from prison with the promise of a return to urban life in a resurrection of the spirit.
The book is not to be read with haste; it is to be savored with the many insights into life in the far north which are to be pondered to be appreciated. Long before the "Gulag Archipeligo" hit the bestseller list this great novel had told the sad and suffering tale of men trapped like mice in a mousetrip of pain.

a great book.......2001-03-11

I can't say that I enjoyed reading this book. It's not the sort of book that you enjoy. I can only say that I'm very glad that I read it. I found it to be both disturbing and compelling.

I can only agree with the other reviewers. This is not a book for everyone. It's not the sort of book that you read for entertainment, for something to do, for the sake of it. But if you want to be challenged and you want to be made to think, you will gain a lot by reading this book.

Not the best of Dostoyevsky's works.......2000-03-07

"Memoirs from the House of the Dead" is the story of Alexander Petrovich, a convict, and it is based on the author's own experiences in 19th century Siberia. Warning: This book was meant to be gloomy, and so it certainly is. The novel is not for those who are looking for entertainment; it is depressing, and, at least in my fancy, the scenery seemed to be grey-tinted at all times... Dialogue is almost non-existent, which makes the book a bit of a difficult read, slow-pacing, sometimes even monotonous.

However, the story does make you think, and Dostoyevsky's books can never be bad - I gave this one only three stars because I am comparing it to his other works and because I think it lacks that certain 'something' that can make even the gloomiest stories curiously captivating - such as "Crime and Punishment," one of Dostoyevsky's best works, which is far from cheerful! "The House of the Dead" is worth reading, but I would not recommend it to someone to whom Dostoyevsky is a fresh acquaintance.

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MEMOIRS FROM THE HOUSE OF THE DEAD
F. M. DOSTOEVSKY
Manufacturer: OXFORD UNIVERSITY PRESS
ProductGroup: Book
Binding: Hardcover
ASIN: B000SDE7VW

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Memoirs From the House of the Dead

Manufacturer: Oxford University Press
ProductGroup: Book
Binding: Hardcover
ASIN: B000GTY358

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Memoirs from the House of the Dead
Fyodor Dostoevsky; Translator Jessie Coulson
Manufacturer: OXFORD UNIVERSITY PRESS
ProductGroup: Book
Binding: Hardcover
ASIN: B000ST71EQ

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