Tensor product algebras, Grassmannians and Khovanov homology

Ben Webster

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arXiv · 1312.7357

Tensor product algebras, Grassmannians and Khovanov homology

Ben Webster

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Abstract

We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan), this has distinct advantage of directly connecting Khovanov homology to a categorification of \$(\mathbb{C}^2)^{\otimes \ell}\$, and admitting a direct generalization to other Lie algebras. While the construction discussed is a special case of that given in previous work of the author, this paper contains new results about the special case of \$\mathfrak{sl}_2\$ showing an explicit connection to Bar-Natan's approach to Khovanov homology, to the geometry of Grassmannians, and to the categorified Jones-Wenzl projectors of Cooper and Krushkal. In particular, we show that the colored Jones homology defined by our approach coincides with that of Cooper and Krushkal.

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Ben Webster. 2013-12-27. Tensor product algebras, Grassmannians and Khovanov homology. https://arxiv.org/abs/1312.7357

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