arXiv subjects
Publications and source records attributed to Cailan Li.
We prove the action of the full twist is a Serre functor in the homotopy category of dihedral Soergel Bimodules. Our proof relies on the representability of a certain partial trace functor, which we prove using the diagrammatics for Hochschild cohomology of Soergel Bimodules. Leveraging these representability results, we then compute the triply-graded link homology of the Whitehead link.
We introduce axioms for towers of infinite-dimensional algebras such that the corresponding Grothendieck groups of projective and finite-dimensional modules are Hopf dual to each other. This duality gives rise to an action of the Hesienberg double -- a generalization of the classical Hesienberg algebra -- on the Grothendieck group. We categorify this action and, as an application, construct a categorical realization of quantum differential operators on $\mathbb{A}^1_{\mathbb{Z}[v,v^{-1}]}$.
We investigate the structure of reduced triply graded link homology $\overline{\mathrm{HHH}}$ in the top/bottom three $T-$degrees for links arising as closures of positive/negative braids. Using a diagrammatic approach to the Hochschild cohomology of Soergel bimodules, we provide explicit computations of $\overline{\mathrm{HHH}}$ as $R-$modules in these degrees. Our results reveal that the homology here is often zero, especially in the negative braid case, and display striking uniformity.
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.