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

Permanent link to this document
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

Subjects
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]

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

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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