Hochschild Cohomology Modular Tensor Categories And Mapping Class Groups I
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Hochschild Cohomology Modular Tensor Categories And Mapping Class Groups I
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Author : Simon Lentner
language : en
Publisher: Springer Nature
Release Date : 2023-07-25
Hochschild Cohomology Modular Tensor Categories And Mapping Class Groups I written by Simon Lentner and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-25 with Science categories.
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.
Hochschild Cohomology Modular Tensor Categories And Mapping Class Groups
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Author : Svea Nora Mierach
language : en
Publisher:
Release Date : 2020
Hochschild Cohomology Modular Tensor Categories And Mapping Class Groups written by Svea Nora Mierach and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with categories.
Hochschild Cohomology Modular Tensor Categories And Mapping Classgroups I General Theory
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Author : Simon Lentner
language : en
Publisher:
Release Date : 2020
Hochschild Cohomology Modular Tensor Categories And Mapping Classgroups I General Theory written by Simon Lentner and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with categories.
Lectures On Tensor Categories And Modular Functors
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Author : Bojko Bakalov
language : en
Publisher: American Mathematical Soc.
Release Date : 2001
Lectures On Tensor Categories And Modular Functors written by Bojko Bakalov and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
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.
Ring Theory 2019 Proceedings Of The Eighth China Japan Korea International Symposium On Ring Theory
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Author : Hideto Asashiba
language : en
Publisher: World Scientific
Release Date : 2021-01-04
Ring Theory 2019 Proceedings Of The Eighth China Japan Korea International Symposium On Ring Theory written by Hideto Asashiba and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-01-04 with Mathematics categories.
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.
Monoidal Categories And The Gerstenhaber Bracket In Hochschild Cohomology
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Author : Reiner Hermann:
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-09-06
Monoidal Categories And The Gerstenhaber Bracket In Hochschild Cohomology written by Reiner Hermann: and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-09-06 with Mathematics categories.
In this monograph, the author extends S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore the author establishes an explicit description of an isomorphism by A. Neeman and V. Retakh, which links Ext-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, the author shows that his construction behaves well with respect to structure preserving functors between exact monoidal categories. The author uses his main result to conclude, that the graded Lie bracket in Hochschild cohomology is an invariant under Morita equivalence. For quasi-triangular bialgebras, he further determines a significant part of the Lie bracket's kernel, and thereby proves a conjecture by L. Menichi. Along the way, the author introduces n-extension closed and entirely extension closed subcategories of abelian categories, and studies some of their properties.
Hochschild Cohomology For Algebras
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Author : Sarah J. Witherspoon
language : en
Publisher: American Mathematical Society
Release Date : 2020-06-30
Hochschild Cohomology For Algebras written by Sarah J. Witherspoon and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-06-30 with Mathematics categories.
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.
Tensor Categories
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Author : Pavel Etingof
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-08-05
Tensor Categories written by Pavel Etingof and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-05 with Mathematics categories.
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.
Twisted Tensor Products Related To The Cohomology Of The Classifying Spaces Of Loop Groups
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Author : Katsuhiko Kuribayashi
language : en
Publisher: American Mathematical Soc.
Release Date : 2006
Twisted Tensor Products Related To The Cohomology Of The Classifying Spaces Of Loop Groups written by Katsuhiko Kuribayashi and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.
Let $G$ be a compact, simply connected, simple Lie group. By applying the notion of a twisted tensor product in the senses of Brown as well as of Hess, we construct an economical injective resolution to compute, as an algebra, the cotorsion product which is the $E_2$-term of the cobar type Eilenberg-Moore spectral sequence converging to the cohomology of classifying space of the loop group $LG$. As an application, the cohomology $H^*(BLSpin(10); \mathbb{Z}/2)$ is explicitly determined as an $H^*(BSpin(10); \mathbb{Z}/2)$-module by using effectively the cobar type spectral sequence and the Hochschild spectral sequence, and further, by analyzing the TV-model for $BSpin(10)$.
Conformal Field Theories And Tensor Categories
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Author : Chengming Bai
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-10-30
Conformal Field Theories And Tensor Categories written by Chengming Bai and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-30 with Mathematics categories.
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.