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On Representation-Finite Gendo-Symmetric Biserial Algebras

Abstract

Gendo-symmetric algebras were introduced by Fang and Koenig (Trans. Amer. Math. Soc., 7:5037–5055, 2016) as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost ν-stable derived equivalences, introduced in Hu and Xi (I. Nagoya Math. J., 200:107–152, 2010), between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost ν-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.

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Acknowledgments

This research was initiated during the “Conference on triangulated categories in algebra, geometry and topology” and “Workshop on Brauer graph algebras” in Stuttgart University, March 2016. We thank Steffen Koenig for comments on an earlier draught. AC is supported by IAR Research Project. Institute for Advanced Research, Nagoya University, and JSPS International Fellowship.

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Correspondence to Aaron Chan.

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Presented by Yuri Drozd.

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Chan, A., Marczinzik, R. On Representation-Finite Gendo-Symmetric Biserial Algebras. Algebr Represent Theor 22, 141–176 (2019). https://doi.org/10.1007/s10468-017-9760-6

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  • DOI: https://doi.org/10.1007/s10468-017-9760-6

Keywords

  • Representation theory of finite dimensional algebras
  • Gorenstein dimension
  • Gendo-symmetric algebra
  • Nakayama algebras
  • Almost ν-stable derived equivalence
  • Brauer tree algebras
  • Dominant dimension

Mathematics Subject Classification 2010

  • Primary 16G10
  • 16E10