论文标题
Dynkin类型的变形前代数$ \ Mathbb {E} _6 $，$ \ Mathbb {E} _7 $和$ \ Mathbb {e} _8 $
An algorithm for the periodicity of deformed preprojective algebras of Dynkin types $\mathbb{E}_6$, $\mathbb{E}_7$ and $\mathbb{E}_8$
论文作者
论文摘要
我们构建了一种数字算法，用于完成一个猜想的证明，该证明声称所有变形的概括性代数的概括性dynkin型都是周期性的。特别是，我们获得了一个算法过程，该过程表明，非平凡的变形前置代数$ \ mathbb {e} _7 $和$ \ Mathbb {e} _8 $仅在特征2中存在于特征2中。 $ \ mathbb {e} _7 $和$ \ mathbb {e} _8 $是周期性的，我们获得了用于分类的算法，直至代数同构。我们通过将猜想减少到与存在合适代数同构的问题的解决方案的解决方案来做到这一点。一个人还表明，我们对猜想的算法方法也适用于广义dynkin类型的网格代数的分类。
We construct a numeric algorithm for completing the proof of a conjecture asserting that all deformed preprojective algebras of generalized Dynkin type are periodic. In particular, we obtain an algorithmic procedure showing that non-trivial deformed preprojective algebras of Dynkin types $\mathbb{E}_7$ and $\mathbb{E}_8$ exist only in characteristic 2. As a consequence, we show that deformed preprojective algebras of Dynkin types $\mathbb{E}_6$, $\mathbb{E}_7$ and $\mathbb{E}_8$ are periodic and we obtain an algorithm for a classification of such algebras, up to algebra isomorphism. We do it by a reduction of the conjecture to a solution of a system of equations associated with the problem of the existence of a suitable algebra isomorphism $φ_f: P^f(\mathbb{E}_n) \to P(\mathbb{E}_n)$ described in Theorem 2.1. One also shows that our algorithmic approach to the conjecture is also applicable to the classification of the mesh algebras of generalized Dynkin type.
