Keith ? - Knitting

Abstract

Talk

Johannes told me, why we are studying a Modul (Algebra) over an Algebra. A modul over an Algebra over a field $K$ is also a vector space over $K$. The left multiplication of the algebra on that vector space induces a linear map on the vector space, giving us a representation.

We’re going to do a few theorems, lemmas but a lot of examples.

4.21

Let $A=KQ/I$, where $I=⟨{\alpha }^2,\gamma \alpha ⟩$. We draw an example with a triangle and one loop. We look at all paths from $1$ to $3$. (The ideal $I$ prevents the paths ${\alpha }^2$ and $\gamma \alpha$ being taken)

We will introduce an algorithm from Shifflers book.

• Compute $P_n$ (the projective modules which start in the vertex $n$)
• Draw arrow $P_i\to P_j$ if $j\to i$
• There’s a third and forth step which I don’t quite get

This algorithm will construct the Auslander-Reiten quiver.

Example

Consider the example $1\leftarrow 2\leftarrow 3\to 4\leftarrow 5$. We now look at all paths that can be drawn from a vertex $i$. We apply the first step. This gives us all projective indecomposable algebras.

Definition

Next, we study $D_n$ a graph with a small fork at the end. Once again, we calculate the projective modules.

Definition

$a:M\to N$ is a source map if

1. $a$ is radical
2. Every radical $g:M\to X$ factors
3. Every $\phi :N\to N$ $\phi a=a$ is isomorphism.

As an example we have $0\to M$ iff $M$ is simple. $M$ is proj. iff $M=P_x$ and $x$ is a sink vertex.

Definition

We define three objects. ${\delta }_{x,y}$ denotes the number of arrows from $x$ to $y$ in the Auslander-Reiten quiver.

Proposition

Let $A$ be a finite dimensional $K$-algebra, $X$ index-$A$-module. … This is a long proposition. What it does is to construct a source map. And somehow it says when an arrow between two indecomposable modules of the AR-quiver exist.