τ-tilting finite triangular matrix algebras

Introduction to τ-tilting theory.

From the viewpoint of mutation, we will give a brief survey of tilting theory and cluster-tilting theory together with a motivation from cluster algebras. Then we will give an introduction to τ-tilting theory which was recently developed.

Derived Equivalent Mates of Triangular Matrix Algebras

A triangular matrix algebra over a field k is defined by a triplet (R, S, M) where R and S are k-algebras and RMS is an SR-bimodule. We show that if R, S and M are finite dimensional and the global dimensions of R and S are finite, then the triangular matrix algebra corresponding to (R, S, M) is derived equivalent to the one corresponding to (S, R, DM), where DM = Homk(M, k) is the dual of M , ...

On (σ, τ)-module extension Banach algebras

Let A be a Banach algebra and X be a Banach A-bimodule. In this paper, we define a new product on $Aoplus X$ and generalize the module extension Banach algebras. We  obtain characterizations of Arens regularity, commutativity, semisimplity, and study the ideal structure and derivations of this new Banach algebra.

Tilting Theory and Cluster Algebras

The purpose of this chapter is to give an introduction to the theory of cluster categories and cluster-tilted algebras, with some background on the theory of cluster algebras, which motivated these topics. We will also discuss some of the interplay between cluster algebras on one side and cluster categories/cluster-tilted algebras on the other, as well as feedback from the latter theory to clus...

Tilting Theory and Quasitilted Algebras