Knot Invariants and Higher Representation Theory.

Cover -- Title page -- Chapter 1. Introduction -- 1. Quantum topology -- 2. Categorification of tensor products -- 3. Topology -- 4. Summary -- Notation -- Acknowledgments -- Chapter 2. Categorification of quantum groups -- 1. Khovanov-Lauda diagrams -- 2. The 2-category -- 3. A spanning set -- 4. Bubble slides -- Chapter 3. Cyclotomic quotients -- 1. A first approach to the categorification of simples -- 2. Categorifications for minimal parabolics -- 2.1. The parabolic categorification -- 2.2. The quiver flag category -- 2.3. The action -- 3. Cyclotomic quotients -- 4. The categorical action on cyclotomic quotients -- 5. Universal categorifications -- Chapter 4. The tensor product algebras -- 1. Stendhal diagrams -- 2. Definition and basic properties -- 3. A basis and spanning set -- 4. Splitting red strands -- 5. The double tensor product algebras -- 6. A Morita equivalence -- 7. Decategorification -- Chapter 5. Standard modules -- 1. Standard modules defined -- 2. Simple modules and crystals -- 3. Stringy triples -- 4. Standard stratification -- 5. Self-dual projectives -- Chapter 6. Braiding functors -- 1. Braiding -- 2. Serre functors -- Chapter 7. Rigidity structures -- 1. Coevaluation and evaluation for a pair of representations -- 2. Ribbon structure -- 3. Coevaluation and quantum trace in general -- Chapter 8. Knot invariants -- 1. Constructing knot and tangle invariants -- 2. The unknot for \fg= ₂ -- 3. Independence of projection -- 4. Functoriality -- Chapter 9. Comparison to category and other knot homologies -- 1. Cyclotomic degenerate Hecke algebras -- 2. Comparison of categories -- 3. The affine case -- 4. Comparison to other knot homologies -- Bibliography -- Back Cover.

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 presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is \mathfrak{sl}_n, the author shows that these categories agree with certain subcategories of parabolic category \mathcal{O} for \mathfrak{gl}_k.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.