November 3, 10, 17 and 24, 14:00-15:30, 7.527
Raymundo Bautista (UNAM Morelia),
Differential Tensor Algebras and their Representations.
1. Matrix Problems, Differential Tensor Algebras and Boxes.
As a motivation we present several classical classification problems of
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
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
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
4. Matrix Bimodule Problems and Algebras.
In this last part we consider matrix bimodule problems considered by several
Drozd, Crawley-Boevey, D. Simson, Y. Zhang. In particular we consider the
problem associated to projective presentations of A-modules with A a finite
dimensional algebra over
some algebraically closed field.