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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 nrepresentation tame algebra is at least n + 2.

We study quivers with potential (QPs) whose Jacobian algebras are finite-dimensional selfinjective. They are an analogue of the 'good QPs' studied by Bocklandt whose Jacobian algebras are 3-Calabi-Yau. We show that 2-representation-finite algebras are truncated Jacobian algebras of selfinjective QPs, which are factor algebras of Jacobian algebras by certain sets of arrows called cuts. We show that selfinjectivity of QPs is preserved under iterated mutation with respect to orbits of the Nakayama permutation. We give a sufficient condition for all truncated Jacobian algebras of a fixed QP to be derived equivalent. We introduce planar QPs which provide us with a rich source of selfinjective QPs.

In this paper, we study n-representation-finite algebras from the viewpoint of the fractionally Calabi-Yau property. We shall show that all n-representation-finite algebras are twisted fractionally Calabi-Yau. We also show that for any > 0, twisted (n( − 1)/ )-Calabi-Yau algebras of global dimension at most n are n-representation-finite. As an application, we give a construction of n-representation-finite algebras using the tensor product.

For each positive integer n we introduce the notion of n-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu. We characterize which n-exangulated categories are n-exact in the sense of Jasso and which are (n + 2)-angulated in the sense of Geiss-Keller-Oppermann. For extriangulated categories with enough projectives and injectives we introduce the notion of n-cluster tilting subcategories and show that under certain conditions such n-cluster tilting subcategories are n-exangulated. (A,C) 7 2.4. Realization of extensions 9 2.5. Definition of n-exangulated categories 11 3. Fundamental properties 14 3.1. Fundamental properties of n-exangulated category 14 3.2. Relative theory 17 4. Typical cases 22 4.1. Extriangulated categories 22 4.2. (n + 2)-angulated categories 24 4.3. n-exact categories 29 5. n-cluster tilting subcategories as n-exangulated categories 41 5.1. Higher extensions in an extriangulated category 41 5.2. The cup product and long exact sequences 44 5.3. From n-cluster tilting subcategories to n-exangulated categories 47 6. Examples 63 6.1. Examples of n-abelian and n-exact categories 64 6.2. Examples of (n + 2)-angulated categories 65 6.3. Examples which are neither n-exact nor (n + 2)-angulated 66 References 70