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n-representation infinite algebras
| Citations: | 4 - 0 self |
BibTeX
@MISC{Herschend_n-representationinfinite,
author = {Martin Herschend and Osamu Iyama and Steffen Oppermann},
title = {n-representation infinite algebras},
year = {}
}
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Abstract
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.
Keyphrases
n-representation infinite algebra n-regular module n-representation tame algebra preprojective algebra new class regular module corresponding preprojective algebra first example tame hereditary algebra classical description important class non-commutative algebraic geometry n-representation infinite representation dimension homological behaviour global dimension dimensional auslander-reiten theory finite dimensional algebra representation infinite hereditary algebra certain analog fano algebra minamoto theory infinite ext 1-orthogonal family general construction finite dimensional module
