Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products

TL;DR

This paper develops a diagrammatic and geometric framework for categorifying tensor products of irreducible representations using 2-quantum groups, with implications for knot invariants and algebraic structures.

Contribution

It introduces an explicit diagrammatic construction of 2-representations of 2-quantum groups that categorify tensor products, generalizing existing algebraic models and establishing their equivalence.

Findings

Categories coincide when both are defined

Proves non-degeneracy of Khovanov-Lauda's 2-category

Shows cyclotomic quiver Hecke algebras are symmetric Frobenius

Abstract

In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying Reshetikhin-Turaev invariants of knots for arbitrary representations, which will be done in a follow-up paper. We consider an algebraic construction of these categories, via an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the quiver Hecke algebra. One of our primary results is that these categories coincide when both are defined. We also investigate finer structure of these categories. Like many similar representation-theoretic 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 Vermas do in more classical representation theory, as test objects for functors. 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.