Raymundo Bautista (UNAM Morelia),

Differential Tensor Algebras and their Representations.

Abstract:

1. Matrix Problems, Differential Tensor Algebras and Boxes.

As a motivation we present several classical classification problems of matrices considered by the Kiev School of Representations of Algebras. Representations of Partially Ordered Sets and Representations of Partially Ordered Sets with Involution. Then we introduce the Representations of Differential Tensor Algebras and show how we can model the above matrix problems with suitable differential tensor algebras. We will see the equivalence between normal boxes and differential tensor algebras.

2. Triangular Differential Tensor Algebras, Exact Structures and Almost Split Sequences.

Here we define layered differential tensor algebras and triangular layered differential tensor algebras, triangular ditalgebras by short. We will see a natural exact structure on the category of representations of triangular ditalgebras. Then under some conditions we prove the existence of almost split sequences in the category of finite-dimensional representations of triangular ditalgebras.

3. Reduction Functors.

Given B a sub differential tensor algebra of A a triangular ditalgebra and X a convenient representation of B, we construct a new triangular ditalgebra A(X), and a functor F(X) from the representations of A(X) into the representations of A. We will see that under certain conditions F(X) is a full and faithful functor. We consider several explicit B and X, among them the ones giving the edge reduction of Kleiner-Roiter and the unravelling of Drozd-(Crawley-Boevey).

4. Matrix Bimodule Problems and Algebras.

In this last part we consider matrix bimodule problems considered by several authors, Drozd, Crawley-Boevey, D. Simson, Y. Zhang. In particular we consider the matrix bimodule problem associated to projective presentations of A-modules with A a finite dimensional algebra over some algebraically closed field.