On wildness of idempotent generated algebras associated with extended Dynkin diagrams

Vitalij M. Bondarenko

Algebra and discrete mathematics, no. 3 (2004), pp. 1-11.

Résumé

Let $\Lambda$ denote an extended Dynkin diagram with vertex set $\Lambda_0=\{0,1,\dots,n\}$. For a vertex $i$, denote by $S(i)$ the set of vertices $j$ such that there is an edge joining $i$ and $j$; one assumes the diagram has a unique vertex $p$, say $p=0$, with $|S(p)|=3$. Further, denote by $\Lambda\setminus 0$ the full subgraph of $\Lambda$ with vertex set $\Lambda_0\setminus\{0\}$. Let $\Delta=(\delta_i\,|\,i\in\Lambda_0)\in\mathbb{Z}^{|\Lambda_0|}$ be an imaginary root of $\Lambda$, and let $k$ be a field of arbitrary characteristic (with unit element 1). We prove that if $\Lambda$ is an extended Dynkin diagram of type $\tilde{D}_4$, $\tilde{E}_6$ or $\tilde{E}_7$, then the $k$-algebra $\mathcal{Q}_k(\Lambda,\Delta)$ with generators $e_i$, $i\in\Lambda_0\setminus\{0\}$, and relations $e_i^2=e_i$, $e_ie_j=0$ if $i$ and $j\ne i$ belong to the same connected component of $\Lambda\setminus 0$, and $\sum_{i=1}^n \delta_i\,e_i=\delta_0 1$ has wild representation type.

Détail

Keywords: idempotent, extended Dynkin diagram, representation, wild typ.