Quantum isometry groups of symmetric groups
Quantum isometry groups of symmetric groups
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对称群的量子等距群
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
P. Sołtan
中科院分区：
文献类型：
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作者：
Jan Liszka;P. Sołtan
我们确定了建立在对称群上的谱三元组的量子等距群，其长度函数是由最近邻转置产生的。事实证明，它们同构于相应对称群的群代数的某些“加倍”。本文在正则乘子Hopf代数的背景下讨论了加倍过程。在最后一节中，我们研究了当n=3时，S_n的等距群对生成元选择的依赖性。我们表明，两种不同的选择的发电机导致非同构的量子同构群，用尽列表的非交换非余交换半单的12维的Hopf代数。这提供了这些Hopf代数的非交换几何解释。
We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of S_n on the choice of generators in the case n=3. We show that two different choices of generators lead to non-isomorphic quantum isometry groups which exhaust the list of non-commutative non-cocommutative semisimple Hopf algebras of dimension 12. This provides non-commutative geometric interpretation of these Hopf algebras.