Hopf coactions on odd spheres

TL;DR

This paper establishes the q-deformed unitary group as the universal quantum symmetry group of the q-odd sphere and identifies it as the quantum isometry group for a specific spectral triple, advancing the understanding of quantum symmetries.

Contribution

It proves the universality of the q-deformed unitary group as the symmetry group of the q-odd sphere and links it to quantum isometries of a spectral triple.

Findings

U_q(N) is the universal quantum group coacting on S_q^{2N-1}

U_q(N) is the quantum isometry group of a natural spectral triple on S_q^{2N-1}

The paper characterizes quantum symmetries of q-deformed spheres

Abstract

We prove that the q-deformed unitary group, i.e., $U_q(N)$, is the universal compact quantum group in the category of (compact) quantum groups which coact on the q-deformed odd sphere $S_q^{2N-1}$ leaving the space spanned by the natural set of generators invariant and preserving the unique $SU_q(N)$ invariant functional on $S_q^{2N-1}$. Using this, we identify $U_q(N)$ as the quantum group of orientation preserving isometries (in the sense of Bhowmick and Goswami \cite{MR2555012}) for a natural spectral triple associated with $S_q^{2N-1}$ constructed by Chakraborty and Pal \cite{MR2458039}.