Algebraic & Geometric Topology

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

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

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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

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

Received: 31 July 2013
Revised: 1 December 2014
Accepted: 14 December 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]

Khovanov homology categorified quantum groups cobordism categories foam categories skew Howe duality link homology


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.

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