On the gendo-symmetric algebra of a trivial extension algebra
On the gendo-symmetric algebra of a trivial extension algebra
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发表时间：
2019
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通讯作者：
T. Honma;T. Aihara;Aaron Chan
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作者：
T. Honma;T. Aihara;Aaron Chan
在代数表示论中，自同态代数起着重要的作用。例如，ProGenerator的自同态代数是与原始代数等价的Morita代数。更一般地，倾斜模的自同态代数是与原始代数等价的。当给定的代数是表示有限的时，模范畴中加法生成元的自同态代数是Auslander代数[A]。因此，自同态代数是一个有趣的研究课题。我们的目的是研究自同态代数的表示类型。然而，在大多数情况下，自同态代数是表示无限的。另一方面，生成元的自同态代数被认为是容易处理的。因此，我们考虑对称代数上生成元的自同态代数，称为广义对称代数[FK]。具体地说，我们的目的是确定一个gendossymm代数何时是表示有限的。在这种情况下，我们还研究了Auslander-Reiten箭图的结构。我们的主要结果可以表述如下。设B是代数A的平凡扩张代数，X是不可分解非投射B-模。考虑由生成元B-ΛX给出的Gendo-对称代数⊕：=EndB(B-⊕X)。在这篇演讲中，我们给出了Λ是表示有限的完整刻画。此外，我们还构造了Λ的稳定的Auslander-Reiten箭图。
In representation theory of algebras, endomorphism algebras play important roles. For example, the endomorphism algebra of a progenerator is Morita equivalent to the original algebra. More generally, the endomorphism algebra of a tilting module is derived equivalence to the original algebra. When a given algebra is representation-finite, the endomorphism algebra of the additive generator in the module category is the Auslander algebra [A]. Thus, endomorphism algebras are interesting subjects of study. Our purpose is to investigate the representation types of endomorphism algebras. However, in most cases, endomorphism algebras are representation-infinite. On the other hand, the endomorphism algebra of a generator is expected to be easy deal with. Therefore, we consider the endomorphism algebra of a generator over a symmetric algebra, so-called a gendo-symmetric algebra [FK]. In particular, our aim is to determine when a gendosymmetric algebra is representation-finite. In the case, we also study the structure of the Auslander-Reiten quiver. Our main result can be stated as follows. Let B be the trivial extension algebra of an algebra A and X an indecomposable non-projective B-module. Consider the gendosymmetric algebra Λ := EndB(B ⊕ X) given by the generator B ⊕ X. In this talk, we give a complete description of Λ being representation-finite. Moreover, we construct the stable Auslander-Reiten quiver of Λ.