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Categorified Quantum Groups and Braid Group Actions


Let $U$ be the quantum group associated to a Cartan matrix indexed by a set $I$. There is a remarkable family $(\theta_i)_{i \in I}$ of symmetries of $U$ giving rise to the action of a braid group. The goal of this dissertation is to explain how to categorify these symmetries. The quantum group is categorified by a certain 2-category $\mathcal U$ introduced by Khovanov-Lauda and Rouquier. We propose two constructions providing a categorification of the symmetry $\theta_i$ on $\mathcal U$.

First, there is a complex $\Theta_i$ in $\mathcal U$. We prove that $\Theta_i$ is invertible in the homotopy category of $\mathcal U$, and that there are homotopy equivalences $\Theta_i E_i \simeq F_i\Theta_i[-1]$, where $E_i,F_i$ denote the Chevalley generators of $\mathcal U$. Hence, conjugation by $\Theta_i$ provides an auto-equivalence of the homotopy category of $\mathcal U$. This auto-equivalence decategorifies to the symmetry $\theta_i$. To obtain these results, we prove a faithfulness property for the 2-representations of $\mathcal U$.

Second, we construct a monoidal functor $T_i$ by defining it on generators and checking relations. We prove that this functor decategorifies to the symmetry $\theta_i$ on a certain subalgebra of the positive part of $U$. Our approach is based on categorifying the formulas defining the symmetry $\theta_i$ on $U$. These formulas involve the adjoint action of $U$, of which we also construct a categorification.

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