The Two-Color Ext Soergel Calculus
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866913851407073280 |
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| author | Li, Cailan |
| author_facet | Li, Cailan |
| contents | We compute Ext groups between Soergel Bimodules associated to the infinite/finite dihedral group for a realization in characteristic 0 and show that they are free right $R-$modules. In particular, we obtain an explicit diagrammatic basis for the Hochschild cohomology of indecomposable Soergel Bimodules. We then give a diagrammatic presentation for the corresponding monoidal category of Ext-enhanced Soergel Bimodules.
As applications, we explicitly compute HOMFLY homology/triply graded link homology $\overline{\mathrm{HHH}}$ for the connect sum of two Hopf links and the negative torus link $T(3,-3)$ as right $R-$modules. Furthermore, we show that the Hochschild cohomology of Soergel Bimodules in finite dihedral type categorifies Gomi's trace, providing a $t-$analog of Soergel's Hom Formula in the dihedral setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_07802 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Two-Color Ext Soergel Calculus Li, Cailan Representation Theory Quantum Algebra We compute Ext groups between Soergel Bimodules associated to the infinite/finite dihedral group for a realization in characteristic 0 and show that they are free right $R-$modules. In particular, we obtain an explicit diagrammatic basis for the Hochschild cohomology of indecomposable Soergel Bimodules. We then give a diagrammatic presentation for the corresponding monoidal category of Ext-enhanced Soergel Bimodules. As applications, we explicitly compute HOMFLY homology/triply graded link homology $\overline{\mathrm{HHH}}$ for the connect sum of two Hopf links and the negative torus link $T(3,-3)$ as right $R-$modules. Furthermore, we show that the Hochschild cohomology of Soergel Bimodules in finite dihedral type categorifies Gomi's trace, providing a $t-$analog of Soergel's Hom Formula in the dihedral setting. |
| title | The Two-Color Ext Soergel Calculus |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2211.07802 |