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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

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.

Khovanov homology, categorified quantum groups, cobordism categories, foam categories, skew Howe duality, link homology
Mathematical Subject Classification 2010
Primary: 81R50
Secondary: 17B37, 57M25, 18G60
Received: 31 July 2013
Revised: 1 December 2014
Accepted: 14 December 2014
Published: 10 December 2015
Aaron D Lauda
Department of Mathematics
University of Southern California
Los Angeles, CA 90089
Hoel Queffelec
Institut Montpelliérain Alexander Grothendieck
Université de Montpellier
34095 Montpellier Cedex 5
David E V Rose
Department of Mathematics
University of Southern California
Los Angeles, CA 90089