关于一次穿孔环面的Jacobi代数
On Jacobian algebras associated with the once-punctured torus
摘要：
我们考虑两个非退化的潜力所产生的一次穿刺环面，这是一个自然的选择，研究和比较：第一个是Labardini潜在的，产生一个有限维雅可比代数，而第二个潜在的产生一个无限维雅可比代数。在本文中，我们确定了这两个雅可比代数的强约化分支图。我们的主要结果是，图是连接在这两种情况下。Plamondon使用一般g-向量参数化了有限维代数的强约化分量。我们证明了有限维Jacobian代数的不可分解强约分量的一般g-向量恰好是Reading确定的一次穿孔环面的泛几何系数。
英文摘要：
We consider two non-degenerate potentials for the quiver arising from the once-punctured torus, which are a natural choice to study and compare: the first is the Labardini-potential, yielding a finite-dimensional Jacobian algebra, whereas the second potential gives rise to an infinite dimensional Jacobian algebra. In this paper we determine the graph of strongly reduced components for both Jacobian algebras. Our main result is that the graph is connected in both cases. Plamondon parametrized the strongly reduced components for finite-dimensional algebras using generic g-vectors. We prove that the generic g-vectors of indecomposable strongly reduced components of the finite-dimensional Jacobian algebra are precisely the universal geometric coefficients for the once-punctured torus, which were determined by Reading.
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