Selfinjective algebras of polynomial growth (Q1107609)

Let k be an algebraically closed field. A Euclidean (resp. Ringel) k- category is a tubular extension B of type (p,q), (2,2,r), (2,3,3), (2,3,4) or (2,3,5) (resp. (3,3,3), (2,4,4), (2,3,6) or (2,2,2,2)) of a tame concealed k-category C. Then \(\hat B\) denotes the repetitive category of B. A group G of k-linear automorphisms of \(\hat B\) is called admissible if it acts freely on the objects of \(\hat B\) and has finitely many orbits. In this case, \(\oplus (\hat B/G)\) denotes the finite dimensional k-algebra associated with the category \^B/G. A finite dimensional k-algebra A is called standard if it admits a simply connected Galois covering. Finally, A is of polynomial growth if there is a natural number m such that the indecomposable finite dimensional A- modules occur, in each dimension \(d\geq 2\), in a finite number of discrete and at most \(d^ m\) one-parameter families. It is shown that a basic, connected, finite dimensional, standard, selfinjective k-algebra A is representation-infinite and of polynomial growth if and only if A is isomorphic to \(\oplus (\hat B/G)\), where B is either a Euclidean or Ringel k-category and G an admissible infinite cyclic group of k-linear automorphisms of \(\hat B.\) Two such algebras \(\oplus (\hat B/G)\) and \(\oplus (\hat B'/G')\) are isomorphic if and only if there exists an isomorphism \(\psi: \hat B\to \hat B'\) such that \(G'=\psi G\psi^{-1}\). In this case, B' is obtained from B by a finite number of reflections. The generators of the admissible infinite cyclic groups G of k-linear automorphisms of \(\hat B\) are also described.