Skip to content
DraconiticResearch
Search Sources API
197,693 studies31 sourcesUpdated 46 min ago
PreprintarXiv licencearXiv

Non-unitarizable representations of compact and discrete quantum groups

  • Fatemeh Khosravi
  • Sang-Gyun Youn
Open at sourcePDFDOICite

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 Gq, as well as to the non-Kac free unitary quantum groups UF+ with F∈GL2(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 𝔽UF and all non-amenable orthogonal free quantum groups 𝔽OF.

Related

  1. Representation stability for compact and discrete quantum groups
    Michael Brannan, Junichiro Matsuda, Erik Séguin
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces
    Sheng Tan
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  3. Compact Quantum Metric Spaces from Weakly Geodesic Length Functions on Hyperbolic Groups
    Are Austad
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  4. Sharp Hausdorff--Young inequalities for some locally compact quantum groups
    Jinsong Wu, Fulin Yang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Approximate Synthetic Spectra of Quantum Systems Associated with Compact Lie Groups
    Junyang Liu, Hang Wang, Houjin Wang +1
    arXivPreprintarXiv licenceFull textAnalysis & PDEsQuantum physics & quantum informationarXiv

Cite

BibTeXRIS
BibTeX
@misc{khosravi2026nonunitarizable,
  title = {{Non-unitarizable representations of compact and discrete quantum groups}},
  author = {Fatemeh Khosravi and Sang-Gyun Youn},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27737},
  eprint = {2608.27737},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27737},
}
RIS
TY  - UNPB
TI  - Non-unitarizable representations of compact and discrete quantum groups
AU  - Fatemeh Khosravi
AU  - Sang-Gyun Youn
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.27737
UR  - https://arxiv.org/abs/2608.27737
AB  - 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∈GL_2(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 𝔽U_F and all non-amenable orthogonal free quantum groups 𝔽O_F.
ER  -