Poisson geometry and representations of PI 4-dimensional Sklyanin algebras
Poisson geometry and representations of PI 4-dimensional Sklyanin algebras
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DOI：
10.1007/s00029-021-00713-7
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发表时间：
2018-02
期刊：
影响因子：
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通讯作者：
Chelsea M. Walton;Xingting Wang;M. Yakimov
中科院分区：
文献类型：
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作者：
Chelsea M. Walton;Xingting Wang;M. Yakimov
TakeS是一个四维Sklyanin(椭圆)代数，它在其中心Z上模有限；因此，Si。我们的第一个结果是构造了一个Poisson Z阶结构，使得Z上的诱导Poisson括号不为零。我们还给出了这个括号的显式Jacobian结构，从而描述了最大谱YofZ的辛核分解。然后，我们结合(1)Poisson序结构的几何，(2)附加于S的椭圆曲线的代数几何方法，以及(3)使用直线和脂肪点模的表示理论方法，对S的不可约表示进行了分类。在此过程中，我们改进了Smith和Tate得到的关于Y的奇异轨迹的刻画的结果。不可约表示的分类结果又被用来确定这些代数的判别式理想的零集。
TakeSto be a 4-dimensional Sklyanin (elliptic) algebra that is module-finite over its centerZ; thus,Sis PI. Our first result is the construction of a PoissonZ-order structure onSsuch that the induced Poisson bracket onZis non-vanishing. We also provide the explicit Jacobian structure of this bracket, leading to a description of the symplectic core decomposition of the maximal spectrumYofZ. We then classify the irreducible representations ofSby combining (1) the geometry of the Poisson order structures, with (2) algebro-geometric methods for the elliptic curve attached toS, along with (3) representation-theoretic methods using line and fat point modules ofS. Along the way, we improve results of Smith and Tate obtaining a description the singular locus ofYfor suchS. The classification results for irreducible representations are in turn used to determine the zero sets of the discriminants ideals of these algebrasS.