On representation dimension of tame cluster tilted algebras

Alfredo Gonzalez Chaio; Sonia Trepode

doi:10.5269/bspm.37626

On representation dimension of tame cluster tilted algebras

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.37626 ◽

2021 ◽

Vol 39 (1) ◽

pp. 107-132

Author(s):

Alfredo Gonzalez Chaio ◽

Sonia Trepode

Keyword(s):

Representation Dimension ◽

Tilted Algebras

The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the weak representation dimension, unfortunately this module is not always a cogenerator. We show for which algebras this module gives the representation dimension.

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Representation dimension of quasi-tilted algebras

Journal of the London Mathematical Society ◽

10.1112/jlms/jdp080 ◽

2010 ◽

Vol 81 (2) ◽

pp. 435-456 ◽

Cited By ~ 8

Author(s):

Steffen Oppermann

Keyword(s):

Representation Dimension ◽

Tilted Algebras

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Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

Algebra Colloquium ◽

10.1142/s1005386713000114 ◽

2013 ◽

Vol 20 (01) ◽

pp. 123-140

Author(s):

Teng Zou ◽

Bin Zhu

Keyword(s):

Simplicial Complex ◽

Root System ◽

Cluster Complex ◽

Cluster Complexes ◽

Endomorphism Algebras ◽

Tilted Algebras ◽

Cluster Categories ◽

Hereditary Algebras ◽

Cluster Tilting ◽

Tilting Objects

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.

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AUSLANDER GENERATORS AND HOMOLOGICAL CONJECTURES

Glasgow Mathematical Journal ◽

10.1017/s0017089513000414 ◽

2013 ◽

Vol 56 (3) ◽

pp. 503-506 ◽

Cited By ~ 1

Author(s):

JIAQUN WEI

Keyword(s):

Artin Algebra ◽

Representation Dimension ◽

Homological Conjectures

AbstractLet A be an artin algebra with representation dimension not more than 3. Assuming that AV is an Auslander generator and M ∈ addAV, we show that both findim(EndAM) and findim(EndAM)op are finite, and consequently the Gorenstein symmetry conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for EndAM.

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