Categorical idempotents via shifted 0-affine algebras (Q6613022)

Let \(A\) be a \(\mathbb{C}\)-algebra with unit \(1_A\). An element \(a \in A\) is an idempotent if \(a^2 = a\). An idempotent \(a \in A\) naturally induces another idempotent \(1_A - a\), which is called its complementary idempotent. In particular, they are mutually orthogonal and give a decomposition of \(1_A\). Y.-H. Hsu shows that a categorical action of shifted 0-affine algebra gives two families of pairs of complementary idempotents in the triangulated monoidal category of triangulated endofunctors for each weight category. He obtains two families of pairs of complementary idempotents in the triangulated monoidal category \(\text{D}^b\text{Coh}(G/P \times G/P)\). As an application, this provides examples where the projection functors of a semiorthogonal decomposition are kernel functors, and the author determines the generators of the component categories in the Grassmannians case.