#### Volume 15, issue 5 (2015)

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

### Aaron D Lauda, Hoel Queffelec and David E V Rose

Algebraic & Geometric Topology 15 (2015) 2517–2608
##### Abstract

We show that Khovanov homology (and its ${\mathfrak{s}\mathfrak{l}}_{3}$ 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 ${\mathfrak{s}\mathfrak{l}}_{m}$ 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 ${\mathfrak{s}\mathfrak{l}}_{3}$ analogs purely from the higher representation theory of categorified quantum groups. In the ${\mathfrak{s}\mathfrak{l}}_{2}$ 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 ${\mathfrak{s}\mathfrak{l}}_{3}$ foam category introduced here.

##### Keywords
Khovanov homology, categorified quantum groups, cobordism categories, foam categories, skew Howe duality, link homology
##### Mathematical Subject Classification 2010
Primary: 81R50
Secondary: 17B37, 57M25, 18G60