We show that the categorical action of the shifted $q=0$ affine algebra can be used to construct semiorthogonal decomposition on the weight categories. The key observation is that the Kapranov exceptional collection can be interpreted as convolutions of Fourier-Mukai kernels by using the categorical action. This result also gives us information about the higher categorical structures (natural transformations) between certain functors in the categorical action.
Exceptional collections, t-structures, and categorical action of shifted $q=0$ affine algebra, $\mathfrak{sl}_{2}$ case
A categorical action of the shifted $q=0$ affine algebra
The 3-point Virasoro algebra and its action on a Fock space
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