Subcategories of fixed points of mutations in root categories with type \(\tilde A _{n-1}\) (Q625946)

Let \(\mathcal{R}(n)\) be the root category of finite-dimensional nilpotent representations of cyclic quiver with \(n\) vertices for \(n\geq 3\). Inspired by \textit{A. Bondal}'s methods [Math. USSR, Izv. 34, No. 1, 23--42 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 1, 25--44 (1989; Zbl 0692.18002)], the authors define mutation with respect to any exceptional object in \(\mathcal{R}(n)\), which is of type \(\tilde{A}_{n-1}\). Then, it is shown that \(\mathcal{R}(n-1)\) is triangle-equivalent to the subcategory of fixed points of certain left (or right) mutation in \(\mathcal{R}(n)\). As an application, an isomorphism of the affine Kac-Moody algebra of type \(\tilde{A}_{n-2}\) to a Lie subalgebra of the Kac-Moody algebra of type \(\tilde{A}_{n-1}\) is deduced.