On the representation type of tensor product algebras

Leszczyński, Zbigniew

doi:10.4064/fm-144-2-143-161

Cited by 44 publications

(26 citation statements)

References 19 publications

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“…With these notations we have an isomorphism of k-algebras kQ /I ⊗ k kQ /I kQ/I (see [12]). However, note that since the natural projections from (Q, I, x) to (Q , I , x ) and (Q , I , x ) are not morphisms of pointed bound quivers, the product quiver is not the product of (Q , I , x ) and (Q , I , x ) in the category of pointed bound quivers.…”

Section: Productsmentioning

confidence: 99%

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Fundamental Groups and Presentations of Algebras

Bustamante

Castonguay

2006

J. Algebra Appl.

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Section: Productsmentioning

confidence: 99%

“…Following [12], we define the product quiver Q = Q ⊗ Q as follows. The vertex set Q 0 is simply Q 0 × Q 0 , whereas the arrow set is…”

Section: Productsmentioning

confidence: 99%

Fundamental Groups and Presentations of Algebras

Bustamante

Castonguay

2006

J. Algebra Appl.

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“…Proof. We know from [11] that if B is a triangular basic algebra then the enveloping algebra B e is also triangular and basic. Then we infer by Theorem 4.2 that A e T ( 1 , 2 )(B e ).…”

Section: Applications To the Enveloping Algebrasmentioning

confidence: 99%

A generalization of trivial extension algebras

Pogorzały¹

2005

Journal of Pure and Applied Algebra

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“…Proof. We know from [11] that A o ⊗ K B is representation-infinite and A e is representation-infinite. But A is a left-right projective A-bimodule which is not projective.…”

Section: 5mentioning

confidence: 99%

“…Then by Theorem 2 all A-bimodules in the whole connected component C of Γ A e containing an indecomposable direct summand of A are left-right projective. Thus lrp(A e ) is representation-infinite, because mod(A e ) is representation-infinite (see [11]). Furthermore, Theorem 1 yields that lrp(A o ⊗ K B) is representation-infinite.…”

Section: 5mentioning

confidence: 99%

Left-right projective bimodules and stable equivalences of Morita type

Pogorzały

2001

Colloq. Math.

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Abstract. We study a connection between left-right projective bimodules and stable equivalences of Morita type for finite-dimensional associative algebras over a field. Some properties of the category of all finite-dimensional left-right projective bimodules for selfinjective algebras are also given.Introduction. Let K be a fixed field. In the representation theory of finite-dimensional associative K-algebras with identity, stable equivalences of Morita type seem to be of particular relevance. They play a substantial role in the representation theory of finite groups (see [7,12,15]). It was observed by Rickard that if two self-injective K-algebras are derived equivalent then they are stably equivalent of Morita type [14].Rickard [14] showed that for two derived equivalent K-algebras their Hochschild cohomology algebras are isomorphic. Happel also proved it but did not publish. This result was generalized in [13] to self-injective Kalgebras A and B which are stably equivalent of Morita type. The main trick used in the proof of this generalization rests heavily on an easy observation that there are two very small subcategories (they consist only of one Ω-orbit) which are stably equivalent; one of them is contained in the category of finite-dimensional A-bimodules, the other in the category of finite-dimensional B-bimodules. An important property of these subcategories is that they consist only of bimodules which are projective as left modules and as right modules. Following Butler and King [8] we call such bimodules left-right projective. They were first studied by Auslander and Reiten [5]. It is natural to ask whether a stable equivalence of Morita type between two finite-dimensional K-algebras A and B induces a stable equivalence between their categories of left-right projective bimodules.One of the main aims of this paper is to give an affirmative answer to the last question. We show even more: existence of a stable equivalence between two finite-dimensional K-algebras is equivalent to existence of a special stable equivalence between the categories of left-right projective bimodules.

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On the representation type of tensor product algebras

Abstract: The representation type of tensor product algebras of finite-dimensional algebras is considered. The characterization of algebras A, B such that A ⊗ B is of tame representation type is given in terms of the Gabriel quivers of the algebras A, B.

Cited by 44 publications

References 19 publications

Fundamental Groups and Presentations of Algebras

Fundamental Groups and Presentations of Algebras

A generalization of trivial extension algebras

Left-right projective bimodules and stable equivalences of Morita type

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