n-Representation infinite algebras

By Martin Herschend, Osamu Iyama, Steffen Oppermann. 2012

105 citations recorded by the source corpus.

From the viewpoint of higher dimensional Auslander-Reiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call n-representation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: n-preprojective, n-preinjective and n-regular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext^1-orthogonal families of modules. Moreover we give general constructions of n-representation infinite algebras. Applying Minamoto's theory on Fano algebras in non-commutative algebraic geometry, we describe the category of n-regular modules in terms of the corresponding preprojective algebra. Then we introduce n-representation tame algebras, and show that the category of n-regular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra. This generalizes the classical description of regular modules over tame hereditary algebras. As an application, we show that the representation dimension of an n-representation tame algebra is at least n+2.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d72b30da9a0f071b44153990f53d810273b47fa4

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Citations in the corpus

Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday (2011, unknown): “In particular A is a (d− 1)-representation-infinite algebra in the sense of [HIO12] and a quasi extremely-Fano algebra in the sense of [MM10].”
n-Abelian and n-exact categories (2014, background): “The class of n-representation-infinite algebras was introduced by Herschend-Iyama-Oppermann in [27] as a higher analog of representation-infinite hereditary algebras from the viewpoint of higher AuslanderReiten theory.”
Auslander-Gorenstein algebras and precluster tilting (2016, background): “) See for example [1, 19, 20, 26, 28, 29, 30, 31, 34, 36] for further results in higher Auslander–Reiten theory.”
An introduction to higher Auslander–Reiten theory (2016, background): “In addition to representation theory [15, 23, 31], it has exhibited connections to commutative algebra, commutative and non-commutative algebraic geometry, and combinatorics, see for example [1, 16, 17, 24, 32].”