Year: 2026
Type: preprint
Abstract: 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 $\mathbb FO_F$.
Source: arXiv (Cornell University)
Authors: Khosravi, Fatemeh, Youn, Sang-Gyun
Institutions: University of Isfahan, Seoul National University
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G45824782, G6390571125, G754601064
OpenAlex ID: W7204831573
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