Automorphism groupoids in noncommutative projective geometry
Automorphism groupoids in noncommutative projective geometry
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DOI：
10.1016/j.jalgebra.2022.03.045
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发表时间：
2018-07
影响因子：
0.9
通讯作者：
Nicholas J Cooney;J. Grabowski
中科院分区：
文献类型：
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作者：
Nicholas J Cooney;J. Grabowski
我们解决了非交换几何中的一个自然问题，即在许多例子中观察到的刚性，由此非交换空间（或等价地它们的坐标代数）与它们的交换对应空间相比具有非常少的自同构。在非交换射影几何的框架中，我们定义了一个广群，它的对象是给定维数的非交换射影空间，并且它的态射对应于这些空间的同构。这个广群是自同构群的自然推广。利用张的工作，我们可以将这种结构转换到代数方面，其中我们考虑非交换射影空间的齐次坐标代数。我们的广群中的态射精确地对应于两个坐标代数的张扭的存在.我们分析了这个自同构广群，使用的点方案的几何，介绍了由Artin-Tate-货车den Bergh，在我们的广群中的态射与点方案的某些自同构.我们将我们的结果应用到两个重要的例子，二维量子射影空间和Sklyanin代数.在这两种情况下，我们都能够使用点方案的几何来完全描述自同构广群的相应分量。这为这些代数的张扭集提供了一个具体的描述。
We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their commutative counterparts.In the framework of noncommutative projective geometry, we define a groupoid whose objects are noncommutative projective spaces of a given dimension and whose morphisms correspond to isomorphisms of these. This groupoid is then a natural generalization of an automorphism group. Using work of Zhang, we may translate this structure to the algebraic side, wherein we consider homogeneous coordinate algebras of noncommutative projective spaces. The morphisms in our groupoid precisely correspond to the existence of a Zhang twist relating the two coordinate algebras.We analyse this automorphism groupoid, using the geometry of the point scheme, as introduced by Artin-Tate-Van den Bergh, to relate morphisms in our groupoid to certain automorphisms of the point scheme.We apply our results to two important examples, the two-dimensional quantum projective space and Sklyanin algebras. In both cases, we are able to use the geometry of the point schemes to fully describe the corresponding component of the automorphism groupoid. This provides a concrete description of the collection of Zhang twists of these algebras.