## KLR algebras and categorification (Spring 2022)

Organized by Cailan Li and Alvaro Martinez. Please email me at alm2297@columbia.edu to join the mailing list.

• Time/date: Mondays at 4.10pm-5.45pm EST.

• Location: Online and in 203 at the Columbia Math Department.

This is a learning seminar on categorification and KLR algebras, a central topic in 2-representation theory. The plan is to spend 7 weeks covering the basics of KLR algebras from the ground up, and then dive into more specialized topics, following the outline below.

Outline of the seminar.

Talks will roughly be 45 minutes followed by a 5 minute break followed by 45 more minutes. Please email me at alm2297@columbia.edu if you'd like to give a talk.

### Rules of the seminar:

• You must have an example/computation in $$\mathfrak{sl}_2$$ or $$\mathfrak{sl}_3$$ in your talk.

• You cannot give a slides talk unless your talk does not have an example/computation in $$\mathfrak{sl}_2$$ or $$\mathfrak{sl}_3$$ .

• Turn your video on (for the most part) as a courtesy to the speaker.

• There are no dumb questions.

If you're giving a talk, please email me the title and abstract by Friday night, and the notes by the day of the talk.

To see the recordings below, the password is: [the password for the zoom meetings]_2022

### Schedule

 Mon Jan 31 Cailan Li Overview of KLR algebras Organizational meeting. Notes Recording Mon Feb 7 Alvaro Martinez KLR Algebras: Essentials We introduce and motivate the construction of the KLR algebras $$R(\nu)$$, with an eye towards the categorification of Lusztig's algebra $$\mathbf{f}$$. We will describe a few examples, paying special attention to the case where $$\Gamma$$ is a single vertex, and $$R(\nu)$$ becomes the NilHecke algebra. A certain faithful representation will be useful to prove some of its properties. At the end, we will generalize this to a faithful representation of $$R(\nu)$$ in general, which will allow us to find bases for KLR algebras. Notes Recording. References: [KL1], [L]. Mon Feb 14 Dinushi Munasinghe KLR Algebras: Properties We go over some properties of KLR algebras $$R(\nu)$$. We find the centre of the algebra, look at the bijection between indecomposable projective and simple modules of $$R(\nu)$$, and end with some isomorphisms between $$R(\nu)$$-modules. Notes Recording Mon Feb 21 Cailan Li KLR Algebras: Categorification We define Induction and Restriction functors between KLR algebras associated to a simply laced quiver $$Q$$ and show they descend to the split Grothendieck group of projectives. We then show that this Grothendieck group has a bialgebra structure and prove that it is in fact isomorphic to the bialgebra $$U_q(\mathfrak{n}_Q^-)$$, the lower half of the quantum group associated to the Cartan datum given by $$Q$$. Time permitting, we will say a few words about the generalization to non-simply laced $$Q$$. Notes Recording References: [KL1], [B]. Mon Feb 28 Dinushi Munasinghe Cellular Algebras We introduce cellular bases and cellular algebras, and then go over some of the properties of these algebras and their modules. Notes Recording Reference: [M]. Mon Mar 7 Pavel Shlykov Integral Cyclotomic Hecke algebras and the Murphy Basis I will define Cyclotomic Hecke algebras, integral Cyclotomic Hecke algebras, define Murphy basis and show that it is cellular. By the theorem from Dinushi's talk this will provide us with a bunch of irreducible modules. The definitions will be in abundance; the proofs won't, but I will try to show some examples. Recording Reference: [M]. Mon Mar 14 at 1.10pm EDT Alvaro Martinez Representations of Cyclotomic KLR algebras We connect our work on integral cyclotomic Hecke algebras $$\mathscr{H}_n^\Lambda$$ to the KLR algebras $$\mathscr{R}_n(\Gamma_e)$$. Namely, we will define certain quotients $$\mathscr{R}_n^\Lambda(\Gamma_e)$$ called cyclotomic KLR algebras, which will turn out to categorify the representation $$L(\Lambda)$$ of $$\widehat{\mathfrak{sl}_e}$$ in future talks. Having upgraded our previous work on cellularity to a graded version, we will prove that $$\mathscr{R}_n^\Lambda(\Gamma_e)$$ is graded cellular in the semisimple case. This will allow us to prove the first instance of the Graded Isomorphism Theorem: $$\mathscr{R}_n^\Lambda(\Gamma_e) \cong \mathscr{H}_n^\Lambda$$. Notes Recording Reference: [M]. Mon Mar 21 Cailan Li Brundan-Kleshchev Graded Isomorphism and Categorification Theorem We will first state the Brundan-Kleshchev Graded Isomorphism Theorem and give the definition of the map in the non semisimple case. For a dominant weight $$\Lambda$$ we define an action of the quantum affine algebra $$U_q(\widehat{sl_e})$$ on the $$\Lambda$$ Fock Space and show that the highest weight integrable module $$L(\Lambda)$$ appears as a submodule. Using cyclotomic Hecke algebras, we then categorify $$L(\Lambda)$$ and show how assuming Ariki's Categorification Theorem, one can show that the canonical basis of $$L(\Lambda)$$ coincides with the basis of self-dual indecomposable projectives of the cyclotomic Hecke algebras. Time permitting, we will then show that a direct consequence will be the following Schur-Weyl dual of the (Kazhdan-)Lusztig conjecture for quantum groups: the decomposition multiplicities of simple modules for the cyclotomic Hecke algebra inside the Specht modules is given by the transition matrix between the canonical basis for $$L(\Lambda)$$ and the standard basis for $$L(\Lambda)$$. Notes Recording Mon Apr 4 Anne Dranowski Diagrammatic and flag 2-categories categorifying Lusztig’s quantum $$sl(2)$$ Define Lusztig’s idempotentized quantum $$sl(2)$$. Survey the diagrammatics of 2-categories. Introduce the intermediate 2-category $$\scr{U}^\ast$$ and establish some properties about it. First, its generating $$1$$-morphisms decompose as expected (i.e. categorifying the usual $$sl(2)$$ relations). Second, it has the right size. Third, it has a 2-representation on the N-flag 2-category categorifying the $$(N+1)$$-dimensional representation of Lusztig’s quantum $$sl(2)$$ Notes Recording Mon Apr 11 Ben Webster Categorification of tensor products I'll review the construction and properties of categorifications of tensor products in the style of Khovanov and Lauda. Notes Recording Thu Apr 21 at 4.10pm Ben Webster Knot homology and categorified tensor products I will discuss applications of the categorifications of tensor products discussed last time to the definition of knot homology, and discuss the relationship to other constructions. Notes Recording Mon Apr 25 Aaron Lauda Skew Howe duality and Knot homology The Reshetikhin-Turaev construction associated knot invariants to the data of a simple Lie algebra and a choice of irreducible representation. The Jones polynomial is the most famous example coming from the Lie algebra sl(2) and its two-dimensional representation. In this talk we will explain Cautis-Kamnitzer-Morrison's novel new approach to studying RT invariants associated to the Lie algebra sl(n). Rather than delving into a morass of representation theory, we will show how two relatively simple Lie theoretic ingredients can be combined with a powerful duality (Howe duality) to give an elementary and diagrammatic construction of these invariants. We will explain how this new framework solved an important open problem in representation theory, proves the q-holonomicity conjecture in knot theory (joint with Garoufalidis and Lê), and leads to a new elementary approach to `categorifying' these knots invariants to link homology theories. Notes Recording Mon May 2 Cailan Li Categorification of Verma modules for quantum $$sl(2)$$ In the first part of the talk we describe and prove various properties about an extension of the nilhecke algebra $$NH_m^\ext$$. In the second part of the talk we explain how to use this extension to categorify universal Verma modules for quantum $$sl(2)$$, and hopefully show that by equipping $$NH_m^\ext$$ with a differential, one can recover the categorification of $$L(q^n)$$, the finite dimensional irreducible representation of $$U_q(sl_2)$$ of highest weight $$q^n$$. Notes Recording