On wildness of idempotent generated algebras associated with extended Dynkin diagrams

Abstract

Let \(\Lambda\) denote an extended Dynkin diagram with vertex set \(\Lambda_0=\{0,1,\ldots, n\}\). 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 \({\cal Q}_k(\Lambda,\Delta)\) 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.