On wildness of idempotent generated algebras associated with extended Dynkin diagrams

Abstract

Let denote an extended Dynkin diagram with vertex set 0 = {0, 1, . . . , 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 \ 0 the full subgraph of with vertex set 0 \ {0}. Let = ( i | i 2 0) 2 Z| 0| be an imaginary root of , and let k be a field of arbitrary characteristic (with unit element 1). We prove that if is an extended Dynkin diagram of type D˜4, E˜6 or E˜7, then the k-algebra Qk( , ) with generators ei, i 2 0 \ {0}, and relations e2 i = ei, eiej = 0 if i and j 6= i belong to the same connected component of \ 0, and Pn i=1 i ei = 01 has wild representation type.