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Knot invariants and higher representation theory

About this Title

Ben Webster, Department of Mathematics, University of Virginia, Charlottesville, Virginia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 250, Number 1191
ISBNs: 978-1-4704-2650-7 (print); 978-1-4704-4206-4 (online)
DOI: https://doi.org/10.1090/memo/1191
Published electronically: September 7, 2017

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Table of Contents

Chapters

• 1. Introduction
• 2. Categorification of quantum groups
• 3. Cyclotomic quotients
• 4. The tensor product algebras
• 5. Standard modules
• 6. Braiding functors
• 7. Rigidity structures
• 8. Knot invariants
• 9. Comparison to category $\mathcal {O}$ and other knot homologies

Abstract

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show 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 $\mathfrak {sl}_n$.

Our 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 $\mathfrak {sl}_n$, we show that these categories agree with certain subcategories of parabolic category $\mathcal {O}$ for $\mathfrak {gl}_k$.

We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory.

The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda’s 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius.

In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.