In my PhD thesis I am studying the Auslander-Reiten theory of a particular class of wild algebras. My question is:

Are there instances of algebras/categories, where the Auslander-Reiten theory of a wild algebra is understood (in the sence that one knows which shapes of components occur) beside the following:

  • Hereditary algebras (Ringel: Finite dimensional hereditary algebras of wild representation type)
  • Canonical algebras and coherent sheaves on a weighted projective line (Lenzing, de la Peña: Wild canonical algebras)
  • Group algebras (Erdmann: On Auslander-Reiten components for group algebras)
  • Local restricted enveloping algebras (Erdmann: The Auslander-Reiten quiver of restricted enveloping algebras)
  • Quantum complete intersections (Bergh, Erdmann: The stable Auslander-Reiten quiver of a quantum complete intersection)

For self-injective Koszul algebras of Loewy length greater than three such that the Yoneda algebra is Noetherian, all (stable) components of the Auslander-Reiten quiver for graded modules are of the form $\mathbb Z A_{\infty}$ (Martinez-Villa, Zacharia: Approximation with modules having linear resolution).

It shouldn't be too hard to find examples where all modules are gradeable, it is more difficult to keep track of graded shifts within each component. The easy way out is to say that each (ungraded) component is of the form $\mathbb Z A_{\infty}/G$ for some group of automorphisms $G$. (Edit: Also known as a tube when $G$ is non-trivial.)


The Auslander-Reiten quiver of Hochschild extension algebras of hereditary algebras can be described in terms of the Auslander-Reiten quiver of the hereditary algebras. This gives examples of such a description for a large class of symmetric algebras. See the book Frobenius algebras II by Skowronski and Yamagata in chapter X.


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