We describe the representation-infinite blocks B of the group algebras KG of finite groups G over algebraically closed fields K for which all simple modules are periodic with respect to the action of the syzygy operators. In particular, we prove that all such blocks B are periodic algebras of period 4. This confirms the periodicity conjecture for blocks of group algebras.
Let Γ be a finite-dimensional hereditary basic algebra. We consider the radical rad Γ as a Γ-bimodule. It is known that there exists a quasi-hereditary algebra 𝓐 such that the category of matrices over rad Γ is equivalent to the category of Δ-filtered 𝓐-modules ℱ(𝓐,Δ). In this note we determine the quasi-hereditary algebra 𝓐 and prove certain properties of its module category.
Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras...
We investigate the structures of Hopf -algebra on the Radford algebras over . All the -structures on are explicitly given. Moreover, these Hopf -algebra structures are classified up to equivalence.
A ring Λ satisfies the Generalized Auslander-Reiten Condition ( ) if for each Λ-module M with for all i > n the projective dimension of M is at most n. We prove that this condition is satisfied by all n-symmetric algebras of quasitilted type.
Let be any rational surface. We construct a tilting bundle on . Moreover, we can choose in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra . The construction starts with a full exceptional sequence of line bundles on and uses universal extensions. If is any smooth projective variety...
A class of finite-dimensional algebras whose Auslander-Reiten quivers have starting but not generalized standard components is investigated. For these components the slices whose slice modules are tilting are considered. Moreover, the endomorphism algebras of tilting slice modules are characterized.