关于紧致度量空间的量子对称性
On quantum symmetries of compact metric spaces
摘要：
紧量子群在紧度量空间$(X,d)$上的作用是（D）-等距的，如果距离函数被$X\times X$上的对角作用保持.我们证明了在这种意义下的等距作用具有以下附加性质：量子群上的概率测度的卷积半群对$X$上的连续函数代数的相应作用收缩Lipschitz常数.换句话说，由于H，它在另一种意义上是等距的。Li，J. Quaegebeur和M. Sabbe;这部分地回答了D.哥斯瓦米我们还介绍了其他可能的等距量子作用的概念，在Wasserstein $p$ -$X$的概率测度之间的距离$p\ge 1$，广泛用于最佳运输。事实证明，所有这些量子等距的定义都符合一个暗示的塔，上面两个在塔的末端。我们推测它们都是等价的。
英文摘要：
An action of a compact quantum group on a compact metric space $(X,d)$ is (D)-isometric if the distance function is preserved by a diagonal action on $X\times X$. We show that an isometric action in this sense has the following additional property: The corresponding action on the algebra of continuous functions on $X$ by the convolution semigroup of probability measures on the quantum group contracts Lipschitz constants. It is, in other words, isometric in another sense due to H. Li, J. Quaegebeur and M. Sabbe; this partially answers a question of D. Goswami. We also introduce other possible notions of isometric quantum action, in terms of the Wasserstein $p$-distances between probability measures on $X$ for $p\ge 1$, used extensively in optimal transportation. It turns out all of these definitions of quantum isometry fit in a tower of implications, with the two above at the extreme ends of the tower. We conjecture that they are all equivalent.
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