$\mathfrak{gl}_n$-webs, categorification and Khovanov-Rozansky homologies
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866917549665419264 |
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| author | Tubbenhauer, Daniel |
| author_facet | Tubbenhauer, Daniel |
| contents | In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality.
Our construction is a $\mathfrak{gl}_n$-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 $H_n(\vec{k})$, and it gives an explicit graded cellular basis of the $2$-hom space between two $\mathfrak{gl}_n$-webs. We use this to give a (in principle) computable version of colored Khovanov-Rozansky $\mathfrak{gl}_n$-link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only $F$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1404_5752 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | $\mathfrak{gl}_n$-webs, categorification and Khovanov-Rozansky homologies Tubbenhauer, Daniel Quantum Algebra Geometric Topology Representation Theory In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a $\mathfrak{gl}_n$-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 $H_n(\vec{k})$, and it gives an explicit graded cellular basis of the $2$-hom space between two $\mathfrak{gl}_n$-webs. We use this to give a (in principle) computable version of colored Khovanov-Rozansky $\mathfrak{gl}_n$-link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only $F$. |
| title | $\mathfrak{gl}_n$-webs, categorification and Khovanov-Rozansky homologies |
| topic | Quantum Algebra Geometric Topology Representation Theory |
| url | https://arxiv.org/abs/1404.5752 |