Quantum deformations of projective three-space
Quantum deformations of projective three-space
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DOI：
10.1016/j.aim.2015.06.005
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发表时间：
2014-03
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影响因子：
--
通讯作者：
Brent Pym
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文献类型：
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作者：
Brent Pym
我们通过展示作为其齐次坐标环的Calabi-Yau代数来描述复射影三维空间可能的非交换变形。我们证明了这类变形的空间参数化有六个不可约分量，并给出了每个族的一般成员在产生子和关系上的显式表示。该证明使用变形量化将问题简化为类似于四维的非模二次泊松结构的分类，我们从Cerveau和Lins Neto在投影空间上的二阶叶状分类中提取。在它们的分类中，与“例外”分量相对应的是投影线的第三对称幂的量子化，它支持经典施瓦岑贝格束的双模量子化。
We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi–Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has exactly six irreducible components, and we give explicit presentations for the generic members of each family in terms of generators and relations. The proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic Poisson structures in four dimensions, which we extract from Cerveau and Lins Neto's classification of degree-two foliations on projective space. Corresponding to the “exceptional” component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical Schwarzenberger bundles.