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Author: Alexander Hoffnung Publisher: ISBN: 9781124121208 Category : Group algebras Languages : en Pages : 124
Book Description
This thesis develops the foundations of the program of groupoidification and presents an application of this program--the Fundamental Theorem of Hecke Operators. In stating this theorem, we develop a theory of enriched bicategories and construct the Hecke bicategory--a categorification of the intertwining operators between permutation representations of a finite group. As an immediate corollary, we obtain a categorification of the Iwahori-Hecke algebra, which leads to solutions of the Zamolodchikov tetrahedron equation. Such solutions are a positive step towards invariants of 2-tangles in 4-dimensional space and constructions of higher-categories with braided structures.
Author: Alexander Hoffnung Publisher: ISBN: 9781124121208 Category : Group algebras Languages : en Pages : 124
Book Description
This thesis develops the foundations of the program of groupoidification and presents an application of this program--the Fundamental Theorem of Hecke Operators. In stating this theorem, we develop a theory of enriched bicategories and construct the Hecke bicategory--a categorification of the intertwining operators between permutation representations of a finite group. As an immediate corollary, we obtain a categorification of the Iwahori-Hecke algebra, which leads to solutions of the Zamolodchikov tetrahedron equation. Such solutions are a positive step towards invariants of 2-tangles in 4-dimensional space and constructions of higher-categories with braided structures.
Author: Anna Beliakova Publisher: American Mathematical Soc. ISBN: 1470424606 Category : Mathematics Languages : en Pages : 376
Book Description
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.
Author: Volodymyr Mazorchuk Publisher: European Mathematical Society ISBN: 9783037191088 Category : Algebraic logic Languages : en Pages : 136
Book Description
The term ``categorification'' was introduced by Louis Crane in 1995 and refers to the process of replacing set-theoretic notions by the corresponding category-theoretic analogues. This text mostly concentrates on algebraical aspects of the theory, presented in the historical perspective, but also contains several topological applications, in particular, an algebraic (or, more precisely, representation-theoretical) approach to categorification. It consists of fifteen sections corresponding to fifteen one-hour lectures given during a Master Class at Aarhus University, Denmark in October 2010. There are some exercises collected at the end of the text and a rather extensive list of references. Video recordings of all (but one) lectures are available from the Master Class website. The book provides an introductory overview of the subject rather than a fully detailed monograph. The emphasis is made on definitions, examples and formulations of the results. Most proofs are either briefly outlined or omitted. However, complete proofs can be found by tracking references. It is assumed that the reader is familiar with the basics of category theory, representation theory, topology, and Lie algebra.
Author: Jacob Greenstein Publisher: Springer Nature ISBN: 3030638499 Category : Mathematics Languages : en Pages : 453
Book Description
This volume collects chapters that examine representation theory as connected with affine Lie algebras and their quantum analogues, in celebration of the impact Vyjayanthi Chari has had on this area. The opening chapters are based on mini-courses given at the conference “Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification”, held on the occasion of Chari’s 60th birthday at the Catholic University of America in Washington D.C., June 2018. The chapters that follow present a broad view of the area, featuring surveys, original research, and an overview of Vyjayanthi Chari’s significant contributions. Written by distinguished experts in representation theory, a range of topics are covered, including: String diagrams and categorification Quantum affine algebras and cluster algebras Steinberg groups for Jordan pairs Dynamical quantum determinants and Pfaffians Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification will be an ideal resource for researchers in the fields of representation theory and mathematical physics.
Author: Jianxun Hu Publisher: Springer Nature ISBN: 9811574510 Category : Mathematics Languages : en Pages : 367
Book Description
This book gathers research papers and surveys on the latest advances in Schubert Calculus, presented at the International Festival in Schubert Calculus, held in Guangzhou, China on November 6–10, 2017. With roots in enumerative geometry and Hilbert's 15th problem, modern Schubert Calculus studies classical and quantum intersection rings on spaces with symmetries, such as flag manifolds. The presence of symmetries leads to particularly rich structures, and it connects Schubert Calculus to many branches of mathematics, including algebraic geometry, combinatorics, representation theory, and theoretical physics. For instance, the study of the quantum cohomology ring of a Grassmann manifold combines all these areas in an organic way. The book is useful for researchers and graduate students interested in Schubert Calculus, and more generally in the study of flag manifolds in relation to algebraic geometry, combinatorics, representation theory and mathematical physics.
Author: Martin Burrow Publisher: Courier Corporation ISBN: 0486145077 Category : Mathematics Languages : en Pages : 210
Book Description
DIVConcise, graduate-level exposition covers representation theory of rings with identity, representation theory of finite groups, more. Exercises. Appendix. 1965 edition. /div
Author: Meinolf Geck Publisher: EPFL Press ISBN: 9780849392436 Category : Mathematics Languages : en Pages : 472
Book Description
After the pioneering work of Brauer in the middle of the 20th century in the area of the representation theory of groups, many entirely new developments have taken place and the field has grown into a very large field of study. This progress, and the remaining open problems (e.g., the conjectures of Alterin, Dade, Broué, James, etc.) have ensured that group representation theory remains a lively area of research. In this book, the leading researchers in the field contribute a chapter in their field of specialty, namely: Broué (Finite reductive groups and spetses); Carlson (Cohomology and representations of finite groups); Geck (Representations of Hecke algebras); Seitz (Topics in algebraic groups); Kessar and Linckelmann (Fusion systems and blocks); Serre (On finite subgroups of Lie groups); Thévenaz (The classification of endo-permutaion modules); and Webb (Representations and cohomology of categories).
Author: Florin Felix Nichita Publisher: MDPI ISBN: 3038973246 Category : Mathematics Languages : en Pages : 239
Book Description
This book is a printed edition of the Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" that was published in Axioms
Author: Alexander Hoffnung
Author: Alexander Hoffnung
Author: Anna Beliakova
Author: Volodymyr Mazorchuk
Author: Jacob Greenstein
Author: Jianxun Hu
Author: Martin Burrow
Author: Meinolf Geck
Author: James Macpherson
Author: Mikhail Khovanov
Author: Florin Felix Nichita