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# Quantum Topology

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**Volume 8, Issue 1, 2017, pp. 113–203**

**DOI: 10.4171/QT/88**

Published online: 2017-03-23

Categorifications of the extended affine Hecke algebra and the affine $q$-Schur algebra $\widehat {\mathbf S} (n,r)$ for $3 \leq r < n$

Marco Mackaay^{[1]}and Anne-Laure Thiel

^{[2]}(1) Universidade do Algarve, Faro, Portugal

(2) University of Uppsala, Sweden

We categorify the extended affine Hecke algebra and the affine quantum Schur algebra $\widehat {\mathbf S} (n,r)$ for $3 \leq r < n$, using results on diagrammatic categorification in affine type A by Elias–Williamson, that extend the work of Elias–Khovanov for finite type A, and Khovanov–Lauda respectively. We also define 2-representations of these categorifications on an extension of the 2-category of affine (singular) Soergel bimodules. These results are the affine analogue of the results in [28].

*Keywords: *Categorification, affine Hecke algebras, quantum algebra

Mackaay Marco, Thiel Anne-Laure: Categorifications of the extended affine Hecke algebra and the affine $q$-Schur algebra $\widehat {\mathbf S} (n,r)$ for $3 \leq r < n$. *Quantum Topol.* 8 (2017), 113-203. doi: 10.4171/QT/88