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Lectures on algebraic categorification. (English) Zbl 1238.18001
The QGM Master Class Series. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-108-8/pbk). ix, 119 p. (2012).
The book presents fifteen lectures given by the author and provides a readable introduction to examples of categorification. One driving force to categorify mathematical structures was the idea of Khovanov who gave a new interpretation of the Jones-Kauffman polynomial in terms of categories. The prototype of categorification can be seen in representing a group as the Grothendieck group of a suitable additive or triangulated category. Pursuing such a program in detail soon leads to the detection of higher structures which, lurking everywhere behind the scene, flavours the subject with excitement – and complication. In the present text, written for master students, the author manages to avoid the technicalities inevitable for any deeper embarkment upon the project.
The first three chapters give a smooth introduction into the basic setup, including a brief account of two-categories and cell structures. The main body of the book deals with the category \(\mathcal{O}\) and its various aspects. It contains a reformulation of the Kazhdan-Lusztig conjecture, provides a categorification of Hecke algebras, cell modules, and similar objects, and contains a few remarks on Koszul duality. Some chapters deal with the quantum group of \(\mathfrak{sl}_2\) and its finite-dimensional representations, Khovanov’s categorification of the Jones polynomial by means of complexes of graded vector spaces, and \(\mathfrak{sl}_2\)-categorification in the sense of Chuang and Rouquier. Broué’s conjecture and Kostant’s problem are discussed in this context. The booklet concludes with a number of exercises and a good selection of literature for further study.

MSC:
18-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
17B55 Homological methods in Lie (super)algebras
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