Algebr Represent Theor (2018) 21:635–681
The Grothendieck Groups and Stable Equivalences
of Mesh Algebras
Received: 30 March 2017 / Accepted: 30 August 2017 / Published online: 7 September 2017
© Springer Science+Business Media B.V. 2017
Abstract We deal with the finite-dimensional mesh algebras given by stable translation
quivers. These algebras are self-injective, and thus the stable module categories have a struc-
ture of triangulated categories. Our main result determines the Grothendieck groups of these
stable module categories. As an application, we give a complete classification of the mesh
algebras up to stable equivalences.
Keywords Translation quivers · Mesh algebras · Grothendieck groups · Stable
Mathematics Subject Classification (2010) Primary 16E20 · Secondary 16D90, 16G10
Let K be a field and Λ be a finite-dimensional K-algebra. The representation theory
of finite-dimensional K-algebras investigates the category of finite-dimensional modules
Λ. One of the useful methods is studying relationships between two finite-dimensional
First, there is an important relationship called derived equivalence, that is, the bounded
) are equivalent as triangulated categories.
Rickard  characterized derived equivalence in terms of tilting complexes. A typi-
cal example of derived equivalences is given by reflections of quivers . Derived
equivalences have been actively studied, see [2, 11, 12, 16], and references therein.
Presented by Vlastimil Dlab.
Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi,
Aichi-ken 464-8602, Japan