Categorical idempotents via shifted 0-affine algebras

TL;DR

This paper explores how categorical actions of shifted 0-affine algebras produce complementary idempotents in triangulated categories, leading to new insights into semiorthogonal decompositions and their generators in algebraic geometry.

Contribution

It introduces a novel connection between shifted 0-affine algebra actions and the construction of complementary idempotents in triangulated categories.

Findings

Identifies two families of complementary idempotents from categorical actions.

Provides examples where projection functors are kernel functors.

Determines generators of component categories in Grassmannians.

Abstract

We show that a categorical action of shifted 0-affine algebra naturally gives two families of pairs of complementary idempotents in the triangulated monoidal category of triangulated endofunctors for each weight category. Consequently, we obtain two families of pairs of complementary idempotents in the triangulated monoidal category $D^bCoh(G/P \times G/P)$. As an application, this provides examples where the projection functors of a semiorthogonal decomposition are kernel functors, and we determine the generators of the component categories in the Grassmannians case.