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Knot Invariants and Higher Representation Theory

Author: Benjamin Thomas Webster

Publisher:

Published: 2017

Total Pages: 141

ISBN-13: 9781470442064

The author constructs knot invariants categorifying the quantum knot variants for all representations of quantum groups. He shows that these invariants coincide with previous invariants defined by Khovanov for \mathfrak{sl}_2 and \mathfrak{sl}_3 and by Mazorchuk-Stroppel and Sussan for \mathfrak{sl}_n. The author's technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presen.

Knot Invariants and Higher Representation Theory

Author: Ben Webster

Publisher: American Mathematical Soc.

Published: 2018-01-16

Total Pages: 141

ISBN-13: 1470426501

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The author constructs knot invariants categorifying the quantum knot variants for all representations of quantum groups. He shows that these invariants coincide with previous invariants defined by Khovanov for sl and sl and by Mazorchuk-Stroppel and Sussan for sl . The author's technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is sl , the author shows that these categories agree with certain subcategories of parabolic category for gl .

Categorification and Higher Representation Theory

Author: Anna Beliakova

Publisher: American Mathematical Soc.

Published: 2017-02-21

Total Pages: 361

ISBN-13: 1470424606

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The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorified representation theory, or higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying actions of algebras on vector spaces where algebra elements act by linear endomorphisms of the vector space, higher representation theory describes the structure present when algebras act on categories, with algebra elements acting by functors. The new level of structure in higher representation theory arises by studying the natural transformations between functors. This enhanced perspective brings into play a powerful new set of tools that deepens our understanding of traditional representation theory. This volume exhibits some of the current trends in higher representation theory and the diverse techniques that are being employed in this field with the aim of showcasing the many applications of higher representation theory. The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.

An Introduction to Quantum and Vassiliev Knot Invariants

Author: David M. Jackson

Publisher: Springer

Published: 2019-05-04

Total Pages: 422

ISBN-13: 3030052133

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This book provides an accessible introduction to knot theory, focussing on Vassiliev invariants, quantum knot invariants constructed via representations of quantum groups, and how these two apparently distinct theories come together through the Kontsevich invariant. Consisting of four parts, the book opens with an introduction to the fundamentals of knot theory, and to knot invariants such as the Jones polynomial. The second part introduces quantum invariants of knots, working constructively from first principles towards the construction of Reshetikhin-Turaev invariants and a description of how these arise through Drinfeld and Jimbo's quantum groups. Its third part offers an introduction to Vassiliev invariants, providing a careful account of how chord diagrams and Jacobi diagrams arise in the theory, and the role that Lie algebras play. The final part of the book introduces the Konstevich invariant. This is a universal quantum invariant and a universal Vassiliev invariant, and brings together these two seemingly different families of knot invariants. The book provides a detailed account of the construction of the Jones polynomial via the quantum groups attached to sl(2), the Vassiliev weight system arising from sl(2), and how these invariants come together through the Kontsevich invariant.

Knot Theory and Its Applications

Author: Krishnendu Gongopadhyay

Publisher: American Mathematical Soc.

Published: 2016-09-21

Total Pages: 357

ISBN-13: 1470422573

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This volume contains the proceedings of the ICTS program Knot Theory and Its Applications (KTH-2013), held from December 10–20, 2013, at IISER Mohali, India. The meeting focused on the broad area of knot theory and its interaction with other disciplines of theoretical science. The program was divided into two parts. The first part was a week-long advanced school which consisted of minicourses. The second part was a discussion meeting that was meant to connect the school to the modern research areas. This volume consists of lecture notes on the topics of the advanced school, as well as surveys and research papers on current topics that connect the lecture notes with cutting-edge research in the broad area of knot theory.

Knots and Physics

Author: Louis H. Kauffman

Publisher: World Scientific

Published: 2013

Total Pages: 865

ISBN-13: 9814383007

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An introduction to knot and link invariants as generalised amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical physics.

Algebraic and Topological Aspects of Representation Theory

Author: Mee Seong Im

Publisher: American Mathematical Society

Published: 2024-01-22

Total Pages: 240

ISBN-13: 1470470349

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This volume contains the proceedings of the virtual AMS Special Session on Geometric and Algebraic Aspects of Quantum Groups and Related Topics, held from November 20–21, 2021. Noncommutative algebras and noncommutative algebraic geometry have been an active field of research for the past several decades, with many important applications in mathematical physics, representation theory, number theory, combinatorics, geometry, low-dimensional topology, and category theory. Papers in this volume contain original research, written by speakers and their collaborators. Many papers also discuss new concepts with detailed examples and current trends with novel and important results, all of which are invaluable contributions to the mathematics community.

Modular Representation Theory Of Finite And P-adic Groups

Author: Wee Teck Gan

Publisher: World Scientific

Published: 2015-02-13

Total Pages: 277

ISBN-13: 9814651826

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This volume is an outgrowth of the program Modular Representation Theory of Finite and p-Adic Groups held at the Institute for Mathematical Sciences at National University of Singapore during the period of 1-26 April 2013. It contains research works in the areas of modular representation theory of p-adic groups and finite groups and their related algebras. The aim of this volume is to provide a bridge — where interactions are rare between researchers from these two areas — by highlighting the latest developments, suggesting potential new research problems, and promoting new collaborations.It is perhaps one of the few volumes, if not only, which treats such a juxtaposition of diverse topics, emphasizing their common core at the heart of Lie theory.

Introduction to Vassiliev Knot Invariants

Author: S. Chmutov

Publisher: Cambridge University Press

Published: 2012-05-24

Total Pages: 521

ISBN-13: 1107020832

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A detailed exposition of the theory with an emphasis on its combinatorial aspects.

Topology and Field Theories

Author: Stephan Stolz

Publisher: American Mathematical Soc.

Published: 2014-04-17

Total Pages: 186

ISBN-13: 147041015X

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This book is a collection of expository articles based on four lecture series presented during the 2012 Notre Dame Summer School in Topology and Field Theories. The four topics covered in this volume are: Construction of a local conformal field theory associated to a compact Lie group, a level and a Frobenius object in the corresponding fusion category; Field theory interpretation of certain polynomial invariants associated to knots and links; Homotopy theoretic construction of far-reaching generalizations of the topological field theories that Dijkgraf and Witten associated to finite groups; and a discussion of the action of the orthogonal group on the full subcategory of an -category consisting of the fully dualizable objects. The expository style of the articles enables non-experts to understand the basic ideas of this wide range of important topics.