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arXiv 1606.06233math.OAmath-phmath.MPmath.QA
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非对易环面的量子对称群

Quantum symmetry groups of noncommutative tori

Michal Banacki, Marcin Marciniak

备注：
25 pages, subsection 4.3 reedited, Propositions 4.2 and 4.4 removed, Remark 5.1 reedited
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摘要：

讨论了紧量子群在Banica和Skalski意义下以保滤方式作用于非对易$n$ -torus $\mathbb{T}_θ^n$代数的必要条件.因此，我们构造了一族紧量子群$\mathbb{G}_θ=(A_θ^n,Δ)$，使得对于每一个$θ$，$\mathbb{G}_θ$是所有紧量子群范畴中的最后一个对象，它们以过滤保持的方式作用于$\mathbb{T}_θ^n$.我们详细描述了C*-代数$A_θ^n$的结构，并提供了一个具体的例子，它的有界算子表示。此外，我们还计算了$\mathbb{G}_θ$的Haar测度.对于$θ=0$，量子群$\mathbb{G}_0$只不过是经典群$\mathbb{T}^n\rtimes S_n$，其中$S_n$是对称群。对于一般的$θ$，$\mathbb{G}_θ$仍然是经典群$\mathbb{T}^n$通过经典群$S_n$的扩展。事实证明，对于$n=2$，代数$A_θ^2$与Hajac和Masuda描述的量子双环面的代数一致。使用一个变化的小子群方法，我们表明，不可约表示的$\mathbb{G}_θ$是在一对一的对应关系与不可约表示的$\mathbb{T}^n\rtimes S_n$。

英文摘要：

We discuss necessary conditions for a compact quantum group to act on the algebra of noncommutative $n$-torus $\mathbb{T}_θ^n$ in a filtration preserving way in the sense of Banica and Skalski. As a result, we construct a family of compact quantum groups $\mathbb{G}_θ=(A_θ^n,Δ)$ such that for each $θ$, $\mathbb{G}_θ$ is the final object in the category of all compact quantum groups acting on $\mathbb{T}_θ^n$ in a filtration preserving way. We describe in details the structure of the C*-algebra $A_θ^n$ and provide a concrete example of its representation in bounded operators. Moreover, we compute the Haar measure of $\mathbb{G}_θ$. For $θ=0$, the quantum group $\mathbb{G}_0$ is nothing but the classical group $\mathbb{T}^n\rtimes S_n$, where $S_n$ is the symmetric group. For general $θ$, $\mathbb{G}_θ$ is still an extension of the classical group $\mathbb{T}^n$ by the classical group $S_n$. It turns out that for $n=2$, the algebra $A_θ^2$ coincides with the algebra of the quantum double-torus described by Hajac and Masuda. Using a variation of the little subgroup method we show that irreducible representations of $\mathbb{G}_θ$ are in one-to-one correspondence with irreducible representations of $\mathbb{T}^n\rtimes S_n$.

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