Local Proof of Algebraic Characterization of Free Actions

Symmetry Integrability and Geometry-Methods and Applications

article

Paul F. Baum¹  Piotr M. Hajac² 

[1] Mathematics Department, The Pennsylvania State University, University Park;Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8;Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski

关键词: compact group;    free action;    Peter–Weyl–Galois condition;   

DOI  :  10.3842/SIGMA.2014.060

来源: National Academy of Science of Ukraine

【 摘 要 】

Let $G$ be a compact Hausdorff topological group acting on a compact Hausdorff topological space $X$. Within the $C^{*}$-algebra $C(X)$ of all continuous complex-valued functions on $X$, there is the Peter-Weyl algebra $\mathcal{P}_G(X)$ which is the (purely algebraic) direct sum of the isotypical components for the action of $G$ on $C(X)$. We prove that the action of $G$ on $X$ is free if and only if the canonical map $\mathcal{P}_G(X)\otimes_{C(X/G)}\mathcal{P}_G(X)\to \mathcal{P}_G(X)\otimes\mathcal{O}(G)$ is bijective. Here both tensor products are purely algebraic, and $\mathcal{O}(G)$ denotes the Hopf algebra of ''polynomial'' functions on $G$.

【 授权许可】

Unknown