## Algebraic & Geometric Topology

### Khovanov homology is a skew Howe $2$–representation of categorified quantum $\mathfrak{sl}_m$

#### Abstract

We show that Khovanov homology (and its $sl3$ variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of $2$–representations of categorified quantum $slm$ via categorical skew Howe duality. Utilizing Cautis–Rozansky categorified clasps we also obtain a unified construction of foam-based categorifications of Jones–Wenzl projectors and their $sl3$ analogs purely from the higher representation theory of categorified quantum groups. In the $sl2$ case, this work reveals the importance of a modified class of foams introduced by Christian Blanchet which in turn suggest a similar modified version of the $sl3$ foam category introduced here.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 5 (2015), 2517-2608.

Dates
Revised: 1 December 2014
Accepted: 14 December 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841026

Digital Object Identifier
doi:10.2140/agt.2015.15.2517

Mathematical Reviews number (MathSciNet)
MR3426687

Zentralblatt MATH identifier
1330.81128

#### Citation

Lauda, Aaron D; Queffelec, Hoel; Rose, David E V. Khovanov homology is a skew Howe $2$–representation of categorified quantum $\mathfrak{sl}_m$. Algebr. Geom. Topol. 15 (2015), no. 5, 2517--2608. doi:10.2140/agt.2015.15.2517. https://projecteuclid.org/euclid.agt/1510841026

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