arXiv subjects
Publications and source records attributed to Yuya Mizuno.
We study silting mutations (Okuyama-Rickard complexes) for selfinjective algebras given by quivers with potential (QPs). We show that silting mutation is compatible with QP mutation. As an application, we get a family of derived equivalences of Jacobian algebras.
Inspired by tau-tilting theory [AIR], we introduce the notion of nu-stable support tau-tilting modules. For any finite dimensional selfinjective algebra Λ, we give bijections between two-term tilting complexes in Kb(proj Λ), nu-stable support tau-tilting Λ-modules and nu-stable functorially finite torsion classes in mod Λ. Moreover, these objects correspond bijectively to selfinjective cluster tilting objects in C if Λ is a 2-CY tilted algebra associated with a Hom-finite 2-CY triangulated category C. As an application, we give a necessary condition such that selfinjective algebras are 2-CY tilted.
We study support $τ$-tilting modules over preprojective algebras of Dynkin type. We classify basic support $τ$-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and other important objects in representation theory. We also study $g$-matrices of support $τ$-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of $g$-matrices and prove that cones given by $g$-matrices coincide with chambers of the associated root systems.
We study the relationship between $n$-cluster tilting modules over $n$ representation finite algebras and the Euler forms. We show that the dimension vectors of cluster-indecomposable modules give the roots of the Euler form. Moreover, we show that cluster-indecomposable modules are uniquely determined by their dimension vectors. This is a generalization of Gabriel's theorem by cluster tilting theory. We call the above roots cluster-roots and investigate their properties. Furthermore, we provide the description of quivers with relations of $n$-APR tilts. Using this, we provide a generalization of BGP reflection functors.
We study the quiver with relations of the endomorphism algebra of an APR tilting module. We give an explicit description of the quiver with relations by graded quivers with potential (QPs) and mutations. The result also implies that mutations of QPs with algebraic cuts induce tilting mutations between the associated truncated Jacobian algebras. Consequently, mutations of QPs provide a rich source of derived equivalence classes of algebras. As an application, we give a sufficient condition of QPs such that Derksen-Weyman-Zelevinsky's question has a positive answer.