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Author: Susan Montgomery Publisher: American Mathematical Soc. ISBN: 0821807382 Category : Mathematics Languages : en Pages : 258
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.
Author: Susan Montgomery Publisher: American Mathematical Soc. ISBN: 0821807382 Category : Mathematics Languages : en Pages : 258
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.
Author: David E. Radford Publisher: World Scientific ISBN: 9814335991 Category : Mathematics Languages : en Pages : 584
Book Description
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.
Author: Y. Bahturin Publisher: Springer Science & Business Media ISBN: 1461302358 Category : Mathematics Languages : en Pages : 240
Book Description
The volume is almost entirely composed of the research and expository papers by the participants of the International Workshop "Groups, Rings, Lie and Hopf Algebras", which was held at the Memorial University of Newfoundland, St. John's, NF, Canada. All four areas from the title of the workshop are covered. In addition, some chapters touch upon the topics, which belong to two or more areas at the same time. Audience: The readership targeted includes researchers, graduate and senior undergraduate students in mathematics and its applications.
Author: Gabriella Böhm Publisher: Springer ISBN: 3319981374 Category : Mathematics Languages : en Pages : 165
Book Description
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.
Author: Publisher: Springer Science & Business Media ISBN: 9781402012204 Category : Mathematics Languages : en Pages : 266
Book Description
The volume is almost entirely composed of the research and expository papers by the participants of the International Workshop "Groups, Rings, Lie and Hopf Algebras", which was held at the Memorial University of Newfoundland, St. John's, NF, Canada. All four areas from the title of the workshop are covered. In addition, some chapters touch upon the topics, which belong to two or more areas at the same time. Audience: The readership targeted includes researchers, graduate and senior undergraduate students in mathematics and its applications.
Author: Stefaan Caenepeel Publisher: CRC Press ISBN: 1000153282 Category : Mathematics Languages : en Pages : 352
Book Description
"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. "
Author: Michiel Hazewinkel Publisher: American Mathematical Soc. ISBN: 0821852620 Category : Mathematics Languages : en Pages : 425
Book Description
Presenting an introduction to the theory of Hopf algebras, the authors also discuss some important aspects of the theory of Lie algebras. This book includes a chapters on the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups, the Hopf algebras of the nonsymmetric and quasisymmetric functions, and the Hopf algebra of permutations.
Author: Jeffrey Bergen Publisher: American Mathematical Soc. ISBN: 1470428059 Category : Mathematics Languages : en Pages : 294
Book Description
This volume contains the proceedings of the International Conference on Groups, Rings, Group Rings, and Hopf Algebras, held October 2–4, 2015 at Loyola University, Chicago, IL, and the AMS Special Session on Groups, Rings, Group Rings, and Hopf Algebras, held October 3–4, 2015, at Loyola University, Chicago, IL. Both conferences were held in honor of Donald S. Passman's 75th Birthday. Centered in the area of group rings and algebras, this volume contains a mixture of cutting edge research topics in group theory, ring theory, algebras and their representations, Hopf algebras and quantum groups.
Author: Susan Montgomery Publisher: American Mathematical Soc. ISBN: 0821850466 Category : Mathematics Languages : en Pages : 290
Book Description
Ring theorists and researchers in invariant theory and operator algebra met at Bowdoin for the 1984 AMS-IMS-SIAM Joint Summer Research Conference to exchange ideas about group actions on rings. This work discusses topics common to the three fields, including: $K$-theory, dual actions, semi-invariants and crossed products.
Author: Justin M. Allman Publisher: ISBN: Category : Languages : en Pages : 112
Book Description
In 1954, Shephard and Todd showed that if $A$ is a polynomial ring and $G$ is a finite group acting as automorphisms on $A$, then the ring of invariants $A^G=\{a\in A : g\cdot a = a, \forall g\in G\}$ is again a polynomial ring exactly when $G$ is generated by reflections. The major goal of this thesis is the computation of several examples en route to a conjecture for an analogous result regarding the ring of invariants for some class of "nice" algebras under finite dimensional Hopf algebra actions. We begin with an introduction to the general study of Hopf algebras and their basic properties, then explain why they are a natural choice to generalize the action of finite groups on rings. We then show that in order to generalize existing theories, we must consider actions of "nontrivial" Hopf algebras, in particular, those that are not isomorphic to group rings or their duals. We compute several examples of such actions, and in particular, we prove that there are no actions of nontrivial semisimple Hopf algebras with dimension less than or equal to 15 on polynomial algebras.