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articleNo Access
${𝔤 𝔩}_{n}$ -webs, categorification and Khovanov–Rozansky homologies
- Daniel Tubbenhauer
Journal of Knot Theory and Its Ramifications01 Oct 2020
Preview Abstract
In this paper, we define an explicit basis for the ${𝔤 𝔩}_{n}$ -web algebra $H_{n} (k)$ (the ${𝔤 𝔩}_{n}$ generalization of Khovanov’s arc algebra) using categorified $q$ -skew Howe duality.
Our construction is a ${𝔤 𝔩}_{n}$ -web version of Hu–Mathas’ graded cellular basis and has two major applications: it gives rise to an explicit isomorphism between a certain idempotent truncation of a thick calculus cyclotomic KLR algebra and $H_{n} (k)$ , and it gives an explicit graded cellular basis of the $2$ -hom space between two ${𝔤 𝔩}_{n}$ -webs. We use this to give a (in principle) computable version of colored Khovanov–Rozansky ${𝔤 𝔩}_{n}$ -link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only $F$ .

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