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ICE QUIVERS WITH POTENTIAL ASSOCIATED WITH TRIANGULATIONS AND COHEN-MACAULAY MODULES OVER ORDERS
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n-representation infinite algebras
"... 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 o ..."
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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.
BIRS 10w5069: Test problems for the theory . . .
, 2010
"... The roots of representation theory go far back into the history of mathematics: the study of symmetry, starting with the Platonic solids and the development of group theory; the study of matrices and the representation theory of groups by Klein, Schur and others which led to the development of the c ..."
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The roots of representation theory go far back into the history of mathematics: the study of symmetry, starting with the Platonic solids and the development of group theory; the study of matrices and the representation theory of groups by Klein, Schur and others which led to the development of the concepts of rings, ideals and modules; the study of normal forms in analysis, in the work of Weierstrass, Jordan and Kronecker, among others; the development of Lie theory. Some of the famous Hilbert’s problems relate representation theory with fundamental geometric concepts. Starting in the middle 60’s of last century, the ‘modern ’ Representation Theory of finite dimensional algebras had a very fast start with three main driving forces: The categorical point of view, represented by Maurice Auslander and his school, leading to the concepts of almost-split sequences, Auslander-Reiten duality, and Auslander-Reiten quivers. The introduction of the concept of quiver representations by Pierre Gabriel, which is now a main tool in the analysis of the representation theory of finite dimensional algebras. The reformulation of problems from representation theory as matrix problems, associated to the Ukrainian school of A. Roiter lead to classification results in certain representation-infinite situations and the conceptual dichotomy of algebras according to their representation type as tame (including representation-finite) or wild. This ‘modern ’ Representation Theory of finite dimensional algebras, typically over an algebraically
