论文标题
连接的局部有限图和离散组量化的量子自动形态组
Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups
论文作者
论文摘要
我们为每个连接的本地有限图$π$构建量子自动形态组$ \ text {qaut} \π$作为本地紧凑型量子组。当$π$是顶点传递的时候，我们将$π$关联到新的统一张量类别$ \ mathcal {c}（π）$，这是我们在$ \ text {qaut} \π$上构建HAAR函数的主要工具。当$π$是有限生成的组的Cayley图时，此统一张量类别是紧凑型量子组的表示类别，其离散二元组可视为基础离散组的规范量化。我们介绍了连接的本地有限图$π$，$π'$的量子同构的几个等效定义，并证明这意味着$ \ text {qaut} \π$和$ \ text {qaut} {qaut} \π'的$。
We construct for every connected locally finite graph $Π$ the quantum automorphism group $\text{QAut}\ Π$ as a locally compact quantum group. When $Π$ is vertex transitive, we associate to $Π$ a new unitary tensor category $\mathcal{C}(Π)$ and this is our main tool to construct the Haar functionals on $\text{QAut}\ Π$. When $Π$ is the Cayley graph of a finitely generated group, this unitary tensor category is the representation category of a compact quantum group whose discrete dual can be viewed as a canonical quantization of the underlying discrete group. We introduce several equivalent definitions of quantum isomorphism of connected locally finite graphs $Π$, $Π'$ and prove that this implies monoidal equivalence of $\text{QAut}\ Π$ and $\text{QAut}\ Π'$.
