Noncommutative geometry based on commutator expansions
Noncommutative geometry based on commutator expansions
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基于换向器展开式的非交换几何
DOI：
10.1515/crll.1998.505.73
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
M. Kapranov
中科院分区：
文献类型：
--
作者：
M. Kapranov
我们在普通交换几何的基础上发展了一种非对易代数几何“微扰状态”的方法。设R是非交换代数，A=R/[R，R]是它的交换化。我们描述了(不存在的)空间Spec(R)中M=Spec(A)的形式邻域。这是一个环空间(M，O)，其中O是M上的某一束非对易环。这样的环空间可以粘合在一起形成更多的全局对象，称为NC-方案。我们特别感兴趣的是NC-流形，即在M的每个点上O的完备化与非对易幂级数的代数(自由结合代数的完备化)同构的NC-流形。利用有序算子的Feynman-Maslov演算，给出了最简单的NC流形-仿射空间的一个显式刻画。我们证明了许多常见的代数簇可以自然地扩展到NC-流形。其中包括所有经典的旗簇和所有向量丛的光滑模空间。
We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal neighborhood of M=Spec(A) in the (nonexistent) space Spec(R). This is a ringed space (M,O) where O is a certain sheaf of noncommutative rings on M. Such ringed spaces can be glued together to form more global objects called NC-schemes. We are especially interested in NC-manifolds, NC-schemes for which the completion of O at every point of M is isomorphic to the algebra of noncommutative power series (completion of the free associative algebra). An explicit description of the simplest NC-manifold, the affine space, is given by using the Feynman-Maslov calculus of ordered operators. We show that many familiar algebraic varieties can be naturally enlarged to NC-manifolds. Among these are all the classical flag varieties and all the smooth moduli spaces of vector bundles.