Grégoire Naisse

• PhD in Mathematics
• None

We construct a categorification of parabolic Verma modules for symmetrizable Kac–Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also...

We explain how Queffelec–Sartori’s construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for gl2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-...

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal \mathfrak{sl}_2 Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorica...

This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. \\ In this part, we categorify all tensor products of Verma modules and integrable modules for quantum $\mathfrak{sl_2}$. The categorificati...

We construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are two-dimensional. This a...

We extend the covering of even and odd Khovanov link homology to tangles, using arc algebras. For this, we develop the theory of quasi-associative algebras and bimodules graded over a category with a 3-cocycle. Furthermore, we show that a covering version of a level 2 cyclotomic half 2-Kac--Moody algebra acts on the bicategory of quasi-associative...

We give a topological description of the two-row Springer fiber over the real numbers. We show its cohomology ring coincides with the oddification of the cohomology ring of the complex Springer fiber introduced by Lauda-Russell. We also realize Ozsv\'ath-Rasmussen-Szab\'o odd TQFT from pullbacks and exceptional pushforwards along inclusion and proj...

We construct a categorification of parabolic Verma modules for all quantum Kac-Moody algebras. The construction relies on a dg-enhancement of cyclotomic Khovanov-Lauda-Rouquier algebras.

We introduce the notion of asymptotic Grothendieck groups for abelian and triangulated categories that are both AB4 and AB4*. We study when the asymptotic Grothendieck group of the heart of a triangulated category with a t-structure is isomorphic to the asymptotic Grothendieck group of the triangulated category itself. We also explain a connexion w...

We construct a categorification of (parabolic) Verma modules for symmetrizable Kac-Moody algebras using KLR-like diagrammatic algebras.

We explain how Queffelec-Sartori's construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for $\mathfrak{gl}_{2n}$. Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov-Rozansky's triply graded link ho...

We study the superalgebra $A_n$ introduced by the authors in a previous paper and use it to construct a 2-Verma module for quantum $\mathfrak{sl}_2$. We prove a uniqueness result about 2-Verma modules on $\Bbbk$-linear 2-categories.

In this paper we study the superalgebra $A_n$, introduced by the authors in previous work on categorification of Verma modules for quantum $\mathfrak{sl}_2$. The superalgebra $A_n$ is akin to the nilHecke algebra, and shares similar properties. In particular, we prove a uniqueness result about 2-Verma modules on $\Bbbk$-linear 2-categories.

We construct an odd version of Khovanov's arc algebra $H^n$. Extending the center to elements that anticommute, we get a subalgebra that is isomorphic to the oddification of the cohomology of the $(n,n)$-Springer varieties. We also prove that the odd arc algebra can be twisted into an associative algebra.

We construct an odd version of Khovanov's arc algebra $H^n$. Extending the center to elements that anticommute, we get a subalgebra that is isomorphic to the oddification of the cohomology of the $(n,n)$-Springer varieties. We also prove that the odd arc algebra can be twisted into an associative algebra.

We give a geometric categorification of the Verma modules $M(\lambda)$ for quantum $\mathfrak{sl}_2$.

We give a geometric categorification of the Verma modules $M(\lambda)$ for quantum $\mathfrak{sl}_2$.

In this master thesis we construct an oddification of the rings $H^n$ from arXiv:math/0103190 using the functor from arXiv:0710.4300 . This leads to a collection of non-associative rings $OH^n_C$ where $C$ represent some choices of signs. Extending the center up to anti-commutative elements, we get a ring $OZ(OH^n_C)$ which is isomorphic to the odd...