The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences

1985 ◽  
Vol 54 (1-2) ◽  
pp. 83-106 ◽  
Author(s):  
Werner Geigle
Keyword(s):  
Representation Theory ◽  
Artin Algebra ◽  
Gabriel Dimension ◽  
Exact Sequences ◽  
Hereditary Artin Algebra

scholarly journals On the Auslander–Reiten quiver of a P1-hereditary artin algebra

2000 ◽  
Vol 151 (1) ◽  
pp. 11-29 ◽  
Author(s):  
Lidia Angeleri Hügel ◽  
Flávio U. Coelho
Keyword(s):  
Artin Algebra ◽  
Hereditary Artin Algebra

On Algebras Stably Equivalent to an Hereditary Artin Algebra

1978 ◽  
Vol 30 (4) ◽  
pp. 817-829 ◽  
Author(s):  
María Inés Platzeck
Keyword(s):  
Finitely Generated ◽  
Artin Algebra ◽  
Finitely Generated Module ◽  
Artin Ring ◽  
Finitely Generated Modules ◽  
Hereditary Artin Algebra

Let Λ be an artin algebra, that is, an artin ring that is a finitely generated module over its center C which is also an artin ring. We denote by mod Λ the category of finitely generated left Λ-modules. We recall that the category of finitely generated modules modulo projectives is the category given by the following data: the objects are the finitely generated Λ-modules.

The Global Dimension of the Endomorphism Algebra of a Generator-cogenerator for a Subcategory

2013 ◽  
Vol 20 (03) ◽  
pp. 443-456
Author(s):  
Jingjing Guo
Keyword(s):  
Global Dimension ◽  
Endomorphism Algebra ◽  
Artin Algebra ◽  
Hereditary Artin Algebra

Let A be a hereditary Artin algebra and T a tilting A-module. The possibilities for the global dimension of the endomorphism algebra of a generator-cogenerator for the subcategory T⊥ in A-mod are determined in terms of relative Auslander-Reiten orbits of indecomposable A-modules in T⊥.

scholarly journals From subcategories to the entire module categories

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Rasool Hafezi
Keyword(s):  
Representation Theory ◽  
Triangular Matrix ◽  
Projective Modules ◽  
Artin Algebra ◽  
Finite Representation ◽  
Triangular Matrix Algebra ◽  
Gorenstein Projective Modules ◽  
Category Of Modules ◽  
Almost Split Sequences ◽  
Auslander Algebra

AbstractIn this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

scholarly journals The representation theory of seam algebras

2020 ◽  
Vol 8 (2) ◽  
Author(s):  
Alexis Langlois-Rémillard ◽  
Yvan Saint-Aubin
Keyword(s):  
Representation Theory ◽  
Boundary Conditions ◽  
Large Class ◽  
Large Family ◽  
Two Dimensional ◽  
Irreducible Modules ◽  
Composition Factors ◽  
Exact Sequences

The boundary seam algebras \mathsf{b}_{n,k}(\beta=q+q^{-1})𝖻n,k(β=q+q−1) were introduced by Morin-Duchesne, Ridout and Rasmussen to formulate algebraically a large class of boundary conditions for two-dimensional statistical loop models. The representation theory of these algebras \mathsf{b}_{n,k}(\beta=q+q^{-1})𝖻n,k(β=q+q−1) is given: their irreducible, standard (cellular) and principal modules are constructed and their structure explicited in terms of their composition factors and of non-split short exact sequences. The dimensions of the irreducible modules and of the radicals of standard ones are also given. The methods proposed here might be applicable to a large family of algebras, for example to those introduced recently by Flores and Peltola, and Crampé and Poulain d’Andecy.

The preprojective algebra of a tame hereditary artin algebra

1987 ◽  
Vol 15 (1-2) ◽  
pp. 425-457 ◽  
Author(s):  
Dagmar Baer ◽  
Werner Geigle ◽  
Helmut Lenzing
Keyword(s):  
Preprojective Algebra ◽  
Artin Algebra ◽  
Hereditary Artin Algebra
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