On wildness of idempotent generated algebras associated with extended Dynkin diagrams

Abstract

Let Λ denote an extended Dynkin diagram with vertex set Λ0 = {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 Λ \ 0 the full subgraph of Λ with vertex set Λ0 \ {0}. Let ∆ = (δi |i ∈ Λ0) ∈ Z |Λ0| be an imaginary root of Λ, and let k be a field of arbitrary characteristic (with unit element 1). We prove that if Λ is an extended Dynkin diagram of type D₄, E₆ or E₇, then the k-algebra Qk(Λ, ∆) with generators ei , i ∈ Λ0 \ {0}, and relations e 2 i = ei , eiej = 0 if i and j 6= i belong to the same connected component of Λ \ 0, and Pn i=1 δi ei = δ01 has wild representation time.