Hochschild Cohomology Modular Tensor Categories And Mapping Class Groups I

Author : Simon Lentner,Svea Nora Mierach,Christoph Schweigert,Yorck Sommerhäuser
Publisher : Springer Nature
Page : 76 pages
File Size : 55,5 Mb
Release : 2023-07-25
Category : Science
ISBN : 9789811946455

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Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups I by Simon Lentner,Svea Nora Mierach,Christoph Schweigert,Yorck Sommerhäuser Pdf

The book addresses a key question in topological field theory and logarithmic conformal field theory: In the case where the underlying modular category is not semisimple, topological field theory appears to suggest that mapping class groups do not only act on the spaces of chiral conformal blocks, which arise from the homomorphism functors in the category, but also act on the spaces that arise from the corresponding derived functors. It is natural to ask whether this is indeed the case. The book carefully approaches this question by first providing a detailed introduction to surfaces and their mapping class groups. Thereafter, it explains how representations of these groups are constructed in topological field theory, using an approach via nets and ribbon graphs. These tools are then used to show that the mapping class groups indeed act on the so-called derived block spaces. Toward the end, the book explains the relation to Hochschild cohomology of Hopf algebras and the modular group.

Author : Hideto Asashiba,Shigeto Kawata,Nanqing Ding,Nam Kyun Kim
Publisher : World Scientific
Page : 256 pages
File Size : 40,9 Mb
Release : 2021-01-04
Category : Mathematics
ISBN : 9789811230301

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Ring Theory 2019 - Proceedings Of The Eighth China-japan-korea International Symposium On Ring Theory by Hideto Asashiba,Shigeto Kawata,Nanqing Ding,Nam Kyun Kim Pdf

Since 1991, the group of ring theorists from China and Japan, joined by Korea from 1995 onwards, took turns to hold the quadrennial international conferences (sometimes also referred to as symposiums). As the proceedings of the eighth conference held in Nagoya, Japan in 2019, this volume consists of a collection of articles by invited speakers (survey) and general speakers (survey and original), all of which were refereed by world experts.The survey articles show the trends of current research and offer clear, thorough explanations that are ideal for researchers also in other specialized areas of ring theory. The original articles display new results, ideas and tools for research investigations in ring theory.The articles cover major areas in ring theory, such as: structures of rings, module theory, homological algebra, groups, Hopf algebras, Lie theory, representation theory of rings, (non-commutative) algebraic geometry, commutative rings (structures, representations), amongst others.This volume is a useful resource for researchers — both beginners and advanced experts — in ring theory.

Author : Bojko Bakalov,Alexander A. Kirillov
Publisher : American Mathematical Soc.
Page : 232 pages
File Size : 53,9 Mb
Release : 2001
Category : Abelian categories
ISBN : 9780821826867

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Lectures on Tensor Categories and Modular Functors by Bojko Bakalov,Alexander A. Kirillov Pdf

This book gives an exposition of the relations among the following three topics: monoidal tensor categories (such as a category of representations of a quantum group), 3-dimensional topological quantum field theory, and 2-dimensional modular functors (which naturally arise in 2-dimensional conformal field theory). The following examples are discussed in detail: the category of representations of a quantum group at a root of unity and the Wess-Zumino-Witten modular functor. The idea that these topics are related first appeared in the physics literature in the study of quantum field theory. Pioneering works of Witten and Moore-Seiberg triggered an avalanche of papers, both physical and mathematical, exploring various aspects of these relations. Upon preparing to lecture on the topic at MIT, however, the authors discovered that the existing literature was difficult and that there were gaps to fill. The text is wholly expository and finely succinct. It gathers results, fills existing gaps, and simplifies some proofs. The book makes an important addition to the existing literature on the topic. It would be suitable as a course text at the advanced-graduate level.

Author : Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 50,5 Mb
Release : 2016-08-05
Category : Algebraic topology
ISBN : 9781470434410

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Tensor Categories by Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik Pdf

Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

Author : David J. Benson,Frederick Ronald Cohen
Publisher : American Mathematical Soc.
Page : 113 pages
File Size : 46,9 Mb
Release : 1991
Category : Complexes
ISBN : 9780821825068

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Mapping Class Groups of Low Genus and Their Cohomology by David J. Benson,Frederick Ronald Cohen Pdf

This series of papers is aimed towards the calculation of the cohomology of the mapping class group of a closed oriented surface of genus two. This is all 2, 3, and 5-torsion. The mod 5 cohomology is given in the first paper, the mod 3 cohomology in the second, and the mod 2 cohomology in the third. Along the way, we investigate many interesting properties of mapping class groups.

Author : Benson Farb,Dan Margalit
Publisher : Princeton University Press
Page : 490 pages
File Size : 44,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9780691147949

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A Primer on Mapping Class Groups by Benson Farb,Dan Margalit Pdf

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Author : Benson Farb
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 54,5 Mb
Release : 2006-09-12
Category : Class groups (Mathematics)
ISBN : 9780821838389

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Problems on Mapping Class Groups and Related Topics by Benson Farb Pdf

The appearance of mapping class groups in mathematics is ubiquitous. The book presents 23 papers containing problems about mapping class groups, the moduli space of Riemann surfaces, Teichmuller geometry, and related areas. Each paper focusses completely on open problems and directions. The problems range in scope from specific computations, to broad programs. The goal is to have a rich source of problems which have been formulated explicitly and accessibly. The book is divided into four parts. Part I contains problems on the combinatorial and (co)homological group-theoretic aspects of mapping class groups, and the way in which these relate to problems in geometry and topology. Part II concentrates on connections with classification problems in 3-manifold theory, the theory of symplectic 4-manifolds, and algebraic geometry. A wide variety of problems, from understanding billiard trajectories to the classification of Kleinian groups, can be reduced to differential and synthetic geometry problems about moduli space. Such problems and connections are discussed in Part III. Mapping class groups are related, both concretely and philosophically, to a number of other groups, such as braid groups, lattices in semisimple Lie groups, and automorphism groups of free groups. Part IV concentrates on problems surrounding these relationships. This book should be of interest to anyone studying geometry, topology, algebraic geometry or infinite groups. It is meant to provide inspiration for everyone from graduate students to senior researchers.

Author : Chengming Bai,Jürgen Fuchs,Yi-Zhi Huang,Liang Kong,Ingo Runkel,Christoph Schweigert
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 45,5 Mb
Release : 2013-10-30
Category : Mathematics
ISBN : 9783642393839

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Conformal Field Theories and Tensor Categories by Chengming Bai,Jürgen Fuchs,Yi-Zhi Huang,Liang Kong,Ingo Runkel,Christoph Schweigert Pdf

The present volume is a collection of seven papers that are either based on the talks presented at the workshop "Conformal field theories and tensor categories" held June 13 to June 17, 2011 at the Beijing International Center for Mathematical Research, Peking University, or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.

Author : John Rognes
Publisher : American Mathematical Soc.
Page : 154 pages
File Size : 40,8 Mb
Release : 2008
Category : Commutative algebra
ISBN : 9780821840764

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Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups by John Rognes Pdf

The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.

Author : Sarah J. Witherspoon
Publisher : American Mathematical Society
Page : 265 pages
File Size : 54,6 Mb
Release : 2020-06-30
Category : Mathematics
ISBN : 9781470462864

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Hochschild Cohomology for Algebras by Sarah J. Witherspoon Pdf

This book gives a thorough and self-contained introduction to the theory of Hochschild cohomology for algebras and includes many examples and exercises. The book then explores Hochschild cohomology as a Gerstenhaber algebra in detail, the notions of smoothness and duality, algebraic deformation theory, infinity structures, support varieties, and connections to Hopf algebra cohomology. Useful homological algebra background is provided in an appendix. The book is designed both as an introduction for advanced graduate students and as a resource for mathematicians who use Hochschild cohomology in their work.

Author : Anonim
Publisher : Unknown
Page : 1884 pages
File Size : 42,7 Mb
Release : 2005
Category : Mathematics
ISBN : UVA:X006195258

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Mathematical Reviews by Anonim Pdf

Author : Gerhard P Hochschild
Publisher : Unknown
Page : 246 pages
File Size : 44,6 Mb
Release : 2019-01-27
Category : Mathematics
ISBN : 4871871622

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The Structure of Lie Groups by Gerhard P Hochschild Pdf

The Structure of Lie Groups presents the basic part of the Lie group theory in a self contained exposition. The main emphasis is put on the use of Lie algebras in dealing with the structural and representation-theoretical features of Lie groups.

Author : Nicolás Andruskiewitsch,Juan Cuadra,Blas Torrecillas
Publisher : American Mathematical Soc.
Page : 347 pages
File Size : 44,9 Mb
Release : 2013-02-21
Category : Mathematics
ISBN : 9780821875643

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Hopf Algebras and Tensor Categories by Nicolás Andruskiewitsch,Juan Cuadra,Blas Torrecillas Pdf

This volume contains the proceedings of the Conference on Hopf Algebras and Tensor Categories, held July 4-8, 2011, at the University of Almeria, Almeria, Spain. The articles in this volume cover a wide variety of topics related to the theory of Hopf algebras and its connections to other areas of mathematics. In particular, this volume contains a survey covering aspects of the classification of fusion categories using Morita equivalence methods, a long comprehensive introduction to Hopf algebras in the category of species, and a summary of the status to date of the classification of Hopf algebras of dimensions up to 100. Among other topics discussed in this volume are a study of normalized class sum and generalized character table for semisimple Hopf algebras, a contribution to the classification program of finite dimensional pointed Hopf algebras, relations to the conjecture of De Concini, Kac, and Procesi on representations of quantum groups at roots of unity, a categorical approach to the Drinfeld double of a braided Hopf algebra via Hopf monads, an overview of Hom-Hopf algebras, and several discussions on the crossed product construction in different settings.

Author : Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts
Publisher : Springer Science & Business Media
Page : 372 pages
File Size : 42,5 Mb
Release : 2003-12-08
Category : Mathematics
ISBN : 3540203575

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Noncommutative Geometry by Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts Pdf

Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.

Author : Carl-Friedrich Bödigheimer,Richard Martin Hain
Publisher : American Mathematical Soc.
Page : 396 pages
File Size : 50,7 Mb
Release : 1993
Category : Mathematics
ISBN : 0821854844

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Mapping Class Groups and Moduli Spaces of Riemann Surfaces by Carl-Friedrich Bödigheimer,Richard Martin Hain Pdf

The study of mapping class groups and moduli spaces of compact Riemann surfaces is currently a central topic in topology, algebraic geometry, and conformal field theory. This book contains proceedings from two workshops held in the summer of 1991, one at the University of G\"ottingen and the other at the University of Washington at Seattle. The papers gathered here represent diverse approaches and contain several important new results. With both research and survey articles, the book appeals to mathematicians and physicists.