紧量子群与离散量子群的不可酉化表示
Non-unitarizable representations of compact and discrete quantum groups
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中文总结 AI 辅助
该研究解决紧与离散量子群表示的相似性问题，开发插值法构造范数近1的不可酉化非退化表示，结合量子提升原理等为多类量子群构建此类表示，完善量子群表示理论相关结论。
中文摘要 AI 辅助
我们研究紧量子群与离散量子群表示的相似性问题。首先证明，紧量子群或离散量子群的每个非退化压缩表示自动是酉的。随后针对紧量子群，我们开发一种插值方法，用于构造范数任意接近1的显式不可酉化非退化表示，该方法适用于对偶具有次指数增长的所有非Kac型紧量子群，包括Drinfeld-Jimbo量子群$G_q$，以及$F\in\operatorname{GL}_2(\mathbb{C})$的非Kac型自由酉量子群$U_F^+$。在离散量子群方面，我们建立经典提升原理的量子类比，用于不可酉化一致有界表示；将该提升原理与关于极大Kac量子子群的最新结果结合，为所有酉自由量子群$\mathbb FU_F$和所有非 amenable 正交自由量子群$\mathbb FO_F$构造出范数任意接近1的显式不可酉化非退化表示。
英文摘要
We study the similarity problem for representations of compact and discrete quantum groups. We first prove that every non-degenerate contractive representation of a compact or discrete quantum group is automatically unitary. Then, for compact quantum groups, we develop an interpolation method for constructing explicit non-unitarizable non-degenerate representations with norms arbitrarily close to $1$. In particular, this applies to all non-Kac compact quantum groups whose dual has subexponential growth, including the Drinfeld--Jimbo quantum groups $G_q$, as well as to the non-Kac free unitary quantum groups $U_F^+$ with $F\in\operatorname{GL}_2(\mathbb{C})$. On the discrete quantum group side, we establish a quantum analogue of the classical lifting principle for non-unitarizable uniformly bounded representations. Combining this lifting principle with recent results on maximal Kac quantum subgroups, we construct explicit non-unitarizable non-degenerate representations with norms arbitrarily close to $1$ for all unitary free quantum groups $\mathbb FU_F$ and all non-amenable orthogonal free quantum groups $\mathbb FO_F$.
