紧度量空间的量子对称性和量子等距性
Quantum symmetries and quantum isometries of compact metric spaces
摘要：
证明了在紧空间上具有忠实作用的具有忠实Haar态的紧量子群必是一个对极有界且对极平方为单位元的Kac代数.证明这一点的主要工具是遍历量子群作用于$C^*$代数的理论。利用这一事实，我们给出了紧度量空间上紧量子群的等距作用的定义，推广了Banica对有限度量空间的定义，并证明了对于某些特殊类的度量空间，在等距作用且比经典等距群“大”的紧量子群的范畴中存在泛对象.
英文摘要：
We prove that a compact quantum group with faithful Haar state which has a faithful action on a compact space must be a Kac algebra, with bounded antipode and the square of the antipode being identity. The main tool in proving this is the theory of ergodic quantum group action on $C^*$ algebras. Using the above fact, we also formulate a definition of isometric action of a compact quantum group on a compact metric space, generalizing the definition given by Banica for finite metric spaces, and prove for certain special class of metric spaces the existence of the universal object in the category of those compact quantum groups which act isometrically and are `bigger' than the classical isometry group.
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