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Backgroud: In his seminal paper A functor-valued invariant of tangles, Khovanov (among many other things) introduced the arc algebra $H^{n}$ and several functors between $H^{n}$ and $H^{m}$ related to tangles. In particular, Khovanov homology of links could be recovered from these functors/bimodules.

Stroppel defined a generalization of Khovanov's arc algebra in Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology by considering arcs with "semi-infinite" ends and the resulting algebra is denoted by $\mathcal{K}^n$. Brundan and Stroppel generalized these algebras further in Highest weight categories arising from Khovanov's diagram algebra I: cellularity, such as $K_{m}^{n}$.

As these papers suggest, the algebras $K_{m}^{n}$ are important for geometric representation theory. Some more recent work shows the connection between such algebras and annular Khovanov homology.

Question: Do we have some concrete computations of Hochschild cohomologies of the algebras $H^n$ and $K_{m}^n$? Certainly we can compute everything out by writing some computer programs, so the question is really about patterns/geometric intepretations of the Hochschild cohomologies in these cases.

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  • $\begingroup$ Is it possible to generalize the technique of Diogo-Lisi and compute symplectic cohomologies instead? $\endgroup$
    – YHBKJ
    Jan 11 '20 at 4:59
  • $\begingroup$ That's possibly true, but it is know that the total rank of Hochschild cohomology of the ordinary arc algebra grows exponentially w.r.t. $n$. To compute the differentials, one needs to compute GW invariants inside some Hilbert scheme, and actually we need some precise information about curves avoiding a divisor. The extended version is some kind of FS category, whose HH* may not have nice "closed-string" cousins. $\endgroup$ Jan 11 '20 at 16:24

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