论文标题
通过颤抖的半神经代数的免费产品表示
Representations of free products of semisimple algebras via quivers
论文作者
论文摘要
令$ \ mathbb {k} $表示一个代数关闭的字段和$ a $ a $ A $ A $ A $ A $ A $ A $ a $ a nemimple Asseciative $ \ mathbb {k} $ - 代数。我们将$ a $ a $ a a $ a有限的无环quiver $γ$关联，并表明有限尺寸$ a $ modules的类别等于类别$ {\ rm rep}（γ）（γ）$的完整子类别为$γ$。在此等价上，对于某些稳定性参数$θ$，简单的$ a $ modules与$γ$的$θ$稳定表示。这为我们提供了必要的条件，即$ a-module是简单的条件，如果模块处于一般位置，这也足够了。即使有不可分解的模块并不简单，我们也证明了一个总位置的模块始终是半imimple的。我们还使用Quivers的nilpotent表示，讨论了任意有限维数模块的构建。最后，当$ \ mathbb {k} $具有特征性零时，我们将结果应用于有限组的免费产品。
Let $\mathbb{K}$ denote an algebraically closed field and $A$ a free product of finitely many semisimple associative $\mathbb{K}$-algebras. We associate to $A$ a finite acyclic quiver $Γ$ and show that the category of finite dimensional $A$-modules is equivalent to a full subcategory of the category ${\rm rep}(Γ)$ of finite dimensional representations of $Γ$. Under this equivalence, the simple $A$-modules correspond exactly to the $θ$-stable representations of $Γ$ for some stability parameter $θ$. This gives us necessary conditions for an $A$-module to be simple, conditions which are also sufficient if the module is in general position. Even though there are indecomposable modules that are not simple, we prove that a module in general position is always semisimple. We also discuss the construction of arbitrary finite dimensional modules using nilpotent representations of quivers. Finally, we apply our results to the case of a free product of finite groups when $\mathbb{K}$ has characteristic zero.
