Abstract
This paper studies self-injective algebras of polynomial growth. We prove that two indecomposable weakly symmetric algebras of domestic type are derived equivalent if and only if they are stably equivalent. Furthermore we prove that for indecomposable weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander–Reiten conjecture for stable equivalences between weakly symmetric algebras of domestic type and for stable equivalences of Morita type between weakly symmetric algebras of polynomial growth.
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G. Zhou was supported by DFG SPP 1388. Part of this work was done during the visit of G. Zhou to the Univerisité de Picardie Jules Verne in March 2010. G. Zhou would like to thank A. Zimmermann for his kind hospitality.
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Zhou, G., Zimmermann, A. Auslander–Reiten conjecture for symmetric algebras of polynomial growth. Beitr Algebra Geom 53, 349–364 (2012). https://doi.org/10.1007/s13366-011-0036-8
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DOI: https://doi.org/10.1007/s13366-011-0036-8
Keywords
- Auslander–Reiten conjecture
- Derived equivalence
- Reynolds ideal
- Self-injective algebra of polynomial growth
- Stable equivalence
- Stable equivalence of Morita type