Non-commutative manifolds, the free square root and symmetric functions in two non-commuting variables
Non-commutative manifolds, the free square root and symmetric functions in two non-commuting variables
复制标题
非交换流形，两个非交换变量中的自由平方根和对称函数
DOI：
10.1112/tlm3.12015
复制
发表时间：
2018
影响因子：
0.8
通讯作者：
Agler J
中科院分区：
文献类型：
--
作者：
Agler J
丰富发展的复流形理论在我们理解多个复变量的全纯函数中发挥着重要作用。很自然地要考虑在几个非交换变量的全纯函数理论中发挥类似作用的流形。在本文中，我们介绍了 nc 流形类，即在每个点都拥有一个邻域的数学对象，该邻域具有维度 nc 宇宙中的 annc 域结构。我们通过构造矩阵平方根函数的非交换黎曼曲面来说明此类流形在自由分析中的使用。第二个例子是构造双变量的初等对称函数的非交换模拟。对于任何对称域，我们构造一个二维非交换流形，使得域上的对称全纯函数与流形上的全纯函数双射对应。我们还推导出两个非对易变量的幂和的经典牛顿-吉拉德公式的一个版本。
The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non‐commuting variables. In this paper we introduce the class ofnc‐manifolds, the mathematical objects that at each point possess a neighborhood that has the structure of annc‐domainin the‐dimensional nc‐universe. We illustrate the use of such manifolds in free analysis through the construction of the non‐commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non‐commutative analog of the elementary symmetric functions in two variables. For any symmetric domain inwe construct a two‐dimensional non‐commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton–Girard formulae for power sums of two non‐commuting variables.