Exceptional collections, t-structures, and categorical action of shifted $q=0$ affine algebra, $\mathfrak{sl}_{2}$ case

TL;DR

This paper demonstrates how the categorical action of the shifted q=0 affine algebra can induce t-structures on weight categories, linking exceptional collections, Fourier-Mukai kernels, and exotic t-structures in algebraic geometry.

Contribution

It introduces a method to construct t-structures from the categorical action of the shifted q=0 affine algebra, connecting Kapranov's exceptional collections with Bezrukavnikov's exotic t-structure.

Findings

Constructed t-structures on weight categories using categorical actions.

Linked Kapranov's exceptional collections with Fourier-Mukai kernels.

Calculated matrix coefficients for algebra generators on Kapranov's basis.

Abstract

In this article, we show that the categorical action of the shifted $q=0$ affine algebra can be used to construct (or induce) t-structures on the weight categories. The main idea is to interpret the exceptional collection constructed by Kapranov as convolution of Fourier-Mukai kernels in the categorical action via using the Borel-Weil-Bott theorem. In particular, when the categories are the bounded derived category of coherent sheaves on Grassmannians. The t-structure we obtain is precisely the exotic t-structure defined by Bezrukavnikov of the exceptional collections given by Kapranov. As an application, we calculate the matrix coefficients for generators of the shifted $q=0$ affine algebra on the basis given by Kapranov exceptional collections.