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**21 - 30**of**30**### 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

### Transfer of stable equivalences of Morita type

, 906

"... Let A and B be finite-dimensional k-algebras over a field k such that A/rad(A) and B/rad(B) are separable. In this note, we consider how to transfer a stable equivalence of Morita type between A and B to that between eAe and fBf, where e and f are idempotent elements in A and in B, respectively. In ..."

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Let A and B be finite-dimensional k-algebras over a field k such that A/rad(A) and B/rad(B) are separable. In this note, we consider how to transfer a stable equivalence of Morita type between A and B to that between eAe and fBf, where e and f are idempotent elements in A and in B, respectively. In particular, if the Auslander algebras of two representation-finite algebras A and B are stably equivalent of Morita type, then A and B themselves are stably equivalent of Morita type. Thus, combining a result with Liu and Xi, we see that two representation-finite algebras A and B over a perfect field are stably equivalent of Morita type if and only if their Auslander algebras are stably equivalent of Morita type. Moreover, since stable equivalence of Morita type preserves n-cluster tilting modules, we extend this result to n-representation-finite algebras and n-Auslander algebras studied by Iyama.

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, 2006

"... We introduce the notion of mutation on the set of n-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules over certain Veronese subrings. 1 ..."

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We introduce the notion of mutation on the set of n-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules over certain Veronese subrings. 1

### n-representation-finite algebras and . . .

"... We introduce the notion of n-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of n-APR tilting, and show that it preserves n-representation-finiteness. We give some combinatorial description of this procedure, and use this to completely ..."

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We introduce the notion of n-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of n-APR tilting, and show that it preserves n-representation-finiteness. We give some combinatorial description of this procedure, and use this to completely describe a class of n-representation-finite algebras called “type A”.

### TORSION CLASSES AND t-STRUCTURES IN HIGHER HOMOLOGICAL ALGEBRA

"... Abstract. Higher homological algebra was introduced by Iyama. It is also known as n-homological algebra where n> 2 is a fixed integer, and it deals with n-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequence ..."

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Abstract. Higher homological algebra was introduced by Iyama. It is also known as n-homological algebra where n> 2 is a fixed integer, and it deals with n-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with n + 2 objects. This was recently formalised by Jasso in the theory of n-abelian categories. There is also a derived version of n-homological algebra, formalised by Geiss, Keller, and Oppermann in the theory of (n+ 2)-angulated categories (the reason for the shift from n to n + 2 is that angulated categories have triangulated categories as the “base case”). We introduce torsion classes and t-structures into the theory of n-abelian and (n + 2)-angulated categories, and prove several results to motivate the definitions. Most of the results concern the n-abelian and (n+2)-angulated categoriesM (Λ) and C (Λ) associated to an n-representation finite algebra Λ, as defined by Iyama and Oppermann. We characterise torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in M (Λ) and intermediate t-structures in C (Λ) which is a category one can reasonably view as the n-derived category of M (Λ). We hint at the link

### 2 RETURNING ARROWS FOR SELF-INJECTIVE ALGEBRAS AND ARTIN-SCHELTER REGULAR ALGEBRAS

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