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Lectures on Algebraic Categorification
Volodymyr Mazorchuk, Uppsala University, Sweden
A publication of the European Mathematical Society.
 The QGM Master Class Series 2012; 128 pp; softcover Volume: 2 ISBN-10: 3-03719-108-2 ISBN-13: 978-3-03719-108-8 List Price: US$36 Member Price: US$28.80 Order Code: EMSQGM/2 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. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in algebra and algebraic geometry. Table of Contents Basics: decategorification and categorification Basics: from categorification of linear maps to $$2$$-categories Basics: $$2$$-representations of finitary $$2$$-categories Category $$\mathcal{O}$$: definitions Category $$\mathcal{O}$$: projective and shuffling functors Category $$\mathcal{O}$$: twisting and completion Category $$\mathcal{O}$$: grading and combinatorics $$\mathbb{S}_n$$-categorification: Soergel bimodules, cells and Specht modules $$\mathbb{S}_n$$-categorification: (induced) cell modules Category $$\mathcal{O}$$: Koszul duality $$\mathfrak{sl}_2$$-categorification: simple finite-dimensional modules Application: categorification of the Jones polynomial $$\mathfrak{sl}_2$$-categorification of Chuang and Rouquier Application: blocks of $$\mathbb{F}[\mathbb{S}_n]$$ and Broué's conjecture Applications of $$\mathbb{S}_n$$-categorifications Exercises Bibliography Index