Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
By Ben Webster. 2010
101 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2f7f53744a1a7eaf2f2e7db84bc0984df9f0e904
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Algebra and Number Theory
- Mikhail Khovanov
- Aaron D. Lauda
- David E. V. Rose
- You Qi
- Hoel Queffelec
- Alexander P. Ellis
- Monica Vazirani
- Ben Webster
In Geometry and Topology
- David E. V. Rose
- Aaron D. Lauda
- Mikhail Khovanov
- Hoel Queffelec
- You Qi
- Alexander P. Ellis
- Monica Vazirani
- Paul Wedrich
In Mathematical Physics
- David E. V. Rose
- Aaron D. Lauda
- Hoel Queffelec
- You Qi
- Mikhail Khovanov
- Alexander P. Ellis
- Monica Vazirani
- Sabin Cautis
Citations in the corpus
- Crystals from categorified quantum groups (2009, methodology): “Before this article went to press, proofs of the full conjecture appeared independently in work of Webster [60] and Kang and Kashiwara [23].”
- Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality (2013, unknown): “Theorem E gives a new instance of based categorifications in the context of category O, but now connecting canonical bases of Hecke algebras with canonical bases of quantum symmetric pairs instead of quantum groups as for instance in [19], [36], [72], [83], see [58] for an overview.”
- Categorifications from planar diagrammatics (2010, background): “Moreover, they serve as a building block in Webster’s categorification of Reshetikhin–Turaev link and tangle invariants [ 49 ,50].”
- An introduction to diagrammatic algebra and categorified quantum sl(2) (2011, background): “The fact that these quotients admit functors lifting the U̇ action appears in [67, 41].”
- The Odd nilHecke Algebra and its Diagrammatics (2011, background): “Various steps in this direction have since been completed, most notably by Webster [18, 19].”
- An approach to categorification of some small quantum groups II (2012, methodology): “e Webster algebras [47] have a diagrammatic description analogous to that of the KLR algebras and their cyclotomic quotients.”
- Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, background): “Recent results of Webster have generalized this statement to arbitrary type [72].”