A maximality result for orthogonal quantum groups

TL;DR

This paper establishes the maximality of the inclusion of the classical orthogonal group into its half-liberated quantum version, showing no intermediate quantum groups exist between them.

Contribution

It proves the maximality of the inclusion $O_n o O_n^*$ for orthogonal quantum groups, combining classical maximality results, isomorphisms, and Hopf algebra techniques.

Findings

No intermediate compact quantum groups between $O_n$ and $O_n^*$

Maximality proven using Lie algebra and matrix methods

Application of a five lemma for cosemisimple Hopf algebras

Abstract

We prove that the quantum group inclusion $O_n \subset O_n^*$ is "maximal", where $O_n$ is the usual orthogonal group and $O_n^*$ is the half-liberated orthogonal quantum group, in the sense that there is no intermediate compact quantum group $O_n\subset G\subset O_n^*$. In order to prove this result, we use: (1) the isomorphism of projective versions $PO_n^*\simeq PU_n$, (2) some maximality results for classical groups, obtained by using Lie algebras and some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.