Discrete Koszul Algebras

This paper is dedicated to the memory of Maurice Auslander, whose love of mathematics will always be an inspiration

The finite dimensional tame hereditary algebras are associated with the extended Dynkin diagrams. An indecomposable module over such an algebra is either preprojective or preinjective or lies in a family of tubes whose tubular type is the corresponding Dynkin diagram. The study of one-point extensions by simple regular modules in such tubes was initiated in [Ri]. We generalise this approach by starting out with algebras which are derived equivalent to a tame hereditary algebra and considering one-point extensions by modules which are simple regular in tubes in the derived category. If the obtained tubular type is again a Dynkin diagram these algebras are called derived Dynkin extensions. Our main theorem says that a representation infinite algebra is derived equivalent to a tame hereditary algebra iff it is an iterated derived Dynkin extension of a tame concealed algebra. As application we get a new proof of a theorem in [AS] about domestic tubular branch enlargements which uses the derived category instead of combinatorial arguments.

A duality theorem for generalized Koszul algebras

Quasitilted One-Point Extensions of Wild Hereditary Algebras

Let A ?= kQ/I be a basic and connected finite dimension algebra over closed field k. In this note show that in case B = A[M] is a tame one-point extension of a tame concealed algebra A by an indecomposable module M, then the trivial extension T(B) = B ? DB is tame if and only if the module M is regular.

On one-point extensions of cluster-tilted algebras

For a generalized one-point extension algebra, it is proved that under certain conditions, each Jordan derivation is the sum of a derivation and an anti-derivation. Moreover, we prove that every Jordan derivation of a dual extension algebra is a derivation.