Representations of regular trees and wild subcategories for generalized Kronecker quivers

Let $r \geq 2$, $k$ be an algebraically closed field of characteristic $p > 0$ and $E_r$ be a $p$-elementary abelian group of rank $r$. It is well known that the category of finite dimensional $kE_r$-modules $\modd kE_r$ is of wild type, whenever $p \geq 3$ or $p = 2$ and $r > 2$. Therefore subclasses with more restrictive properties have been studied. In 2008 Carlson, Friedlander and Suslin introduced the subclass of modules of constant rank and modules with even more restrictive properties, called equal images property and equal kernels property. In her PhD thesis Worch uses a faithful exact functor $\fF_{n} \colon \modd B(n,r) \to \modd kE_r$ from the module category of the generalized Beilinson algebra $B(n,r)$ on $n$ vertices into the module category of the group algebra $kE_r$ to define the equal images and equal kernels property in the framework of the Beilinson algebra. We follow this approach in the special case $n = 2$ and $r \geq 3$, i.e. $B(n,r)$ is the generalized Kronecker $\cK_r = k\Gamma_r$ algebra with $r \geq 3$ arrows. Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules $E_r$, we introduce the notions of modules with constant $d$-radical rank and modules with constant $d$-socle rank for $\cK_r$ and $1 \leq d \leq r-1$. We study subcategories given by modules with the equal $d$-radical property and the equal $d$-socle property. Utilizing the Simplification method due to Ringel, we prove that these subcategories in $\modd \cK_r$ are of wild type. Then we use the functor $\fF_{2} \colon \modd \cK_r \to \modd kE_r$ to transfer our results to $\modd kE_r$. This thesis is also motivated by the aim to give new insights into the wild category $\modd \cK_r$.


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