Sharp hypercontractivity for free orthogonal quantum groups of Kac type
Abstract
Let N≥2, let F∈ GLN(ℂ), and let OF+ be the associated free orthogonal quantum group of Kac type, with tracial Haar state h. We prove that, for 1<p≤ r<∞, the normalized heat semigroup Pt=e−tL on OF+ \cite{CFK14,BVY21} is hypercontractive with the optimal time |Pt:Lp(OF+)→ Lr(OF+)|≤1 Longleftrightarrow t≥1/2log(r−1)/(p−1). In particular, this solves a conjecture of Brannan, Vergnioux and Youn \cite{BVY21}. To prove this hypercontractivity result, we establish the equivalent logarithmic Sobolev inequality Ent(|x|2)≤2E(x,x), x∈ D(E), with sharp constant $2$, where Ent(y)=h(ylog y)−h(y)log h(y) is the relative entropy and E(x,x)=|L1/2x|22 is the Dirichlet form associated with Pt. The main ingredients are the cubic-majorant idea from the recent work of Frank--Ivanisvili \cite{FI26} and Xie--Zhang \cite{XZC26,XZ26}, and a centered third-moment estimate coming from the representation theory. We also prove the sharp Beckner inequalities (|x|p2−|x|22)/(p−2)≤E(x,x), 1≤ p≤3, p≠2, whose limit as p→2 recovers the logarithmic Sobolev inequality above. Moreover, when N=2, the results can be sharpened by replacing the heat semigroup with Pt=e−tΛ, where Λ=√(I+3L)−I. The key step is to use the graded-twist realization of SU−1(2) to transfer Beckner's sharp p=3 inequality on SU(2) \cite{Beckner93}.
Related
- Sharp logarithmic Sobolev inequalities on free groups by variational comparison
- Sharp Hausdorff--Young inequalities for some locally compact quantum groups
- Quantum Entropy Contraction and Factorization from Hypercontractivity
- Hypercontractivity and entropy decay for infinite-dimensional quantum Gibbs samplers
- Quantum Reverse Hypercontractivity for KMS-Symmetric Quantum Markov Semigroups
BibTeX
@misc{zhang2026sharp,
title = {{Sharp hypercontractivity for free orthogonal quantum groups of Kac type}},
author = {Haonan Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.29694},
eprint = {2606.29694},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.29694},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Sharp hypercontractivity for free orthogonal quantum groups of Kac type
AU - Haonan Zhang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2606.29694
UR - https://arxiv.org/abs/2606.29694
AB - Let N≥2, let F∈ GL_N(ℂ), and let O_F^+ be the associated free orthogonal quantum group of Kac type, with tracial Haar state h. We prove that, for 1<p≤ r<∞, the normalized heat semigroup P_t=e^−tL on O_F^+ \cite{CFK14,BVY21} is hypercontractive with the optimal time |P_t:L_p(O_F^+)→ L_r(O_F^+)|≤1 Longleftrightarrow t≥1/2log(r−1)/(p−1). In particular, this solves a conjecture of Brannan, Vergnioux and Youn \cite{BVY21}. To prove this hypercontractivity result, we establish the equivalent logarithmic Sobolev inequality Ent(|x|^2)≤2E(x,x), x∈ D(E), with sharp constant $2$, where Ent(y)=h(ylog y)−h(y)log h(y) is the relative entropy and E(x,x)=|L^1/2x|_2^2 is the Dirichlet form associated with P_t. The main ingredients are the cubic-majorant idea from the recent work of Frank--Ivanisvili \cite{FI26} and Xie--Zhang \cite{XZC26,XZ26}, and a centered third-moment estimate coming from the representation theory. We also prove the sharp Beckner inequalities (|x|_p^2−|x|_2^2)/(p−2)≤E(x,x), 1≤ p≤3, p≠2, whose limit as p→2 recovers the logarithmic Sobolev inequality above. Moreover, when N=2, the results can be sharpened by replacing the heat semigroup with P_t=e^−tΛ, where Λ=√(I+3L)−I. The key step is to use the graded-twist realization of SU_−1(2) to transfer Beckner's sharp p=3 inequality on SU(2) \cite{Beckner93}.
ER -