Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras
Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras
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DOI：
10.1007/s00208-022-02435-3
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发表时间：
2018-03
影响因子：
1.4
通讯作者：
Z. Hua;Gui-song Zhou
中科院分区：
文献类型：
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作者：
Z. Hua;Gui-song Zhou
在本文中，我们证明了对于一个有限的quiverq，在假定Jacobi代数是有限维的情况下，由它的Jacobi代数和它的第0次Hochschild同调中由势表示的类决定了其完全路径代数的变量形式变化的势的等价类。这是关于孤立超曲面奇点的著名的Mather和Yau定理的一个非交换类似。我们还证明了在雅可比代数是有限维的情况下，势的正确等价类是由其足够高的射流决定的。这两个定理可以看作是非交换幂级数奇点理论的第一步。作为一个应用，我们证明了如果Jacobi代数是有限维的，那么对应的完全Ginzburg代数及其（拓扑）广义聚类是由Jacobi代数的同构类型及其0次Hochschild同构中由势表示的类决定的。
In this article, we prove that for a finite quiverQthe equivalence class of a potential up to formal change of variables of the complete path algebra, is determined by its Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential assuming the Jacobi algebra is finite dimensional. This is a noncommutative analogue of the famous theorem of Mather and Yau on isolated hypersurface singularities. We also prove that the right equivalence class of a potential is determined by its sufficiently high jet assuming the Jacobi algebra is finite dimensional. These two theorems can be viewed as a first step towards the singularity theory of noncommutative power series. As an application, we show that if the Jacobi algebra is finite dimensional then the corresponding complete Ginzburg dg-algebra, and the (topological) generalized cluster category thereof, are determined by the isomorphic type of the Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential.