Compact Quantum Group Extensions of $USp_q(2n)$, $O_q(n)$ and $SO_q(2n)$

TL;DR

This paper constructs new compact quantum group extensions for q-deformed classical groups, adding a central unitary element to extend existing quantum groups like USp_q(2n), O_q(n), and SO_q(2n).

Contribution

It introduces a method to extend standard compact quantum groups with a central unitary element, creating new quantum group structures.

Findings

Constructed compact quantum group extensions with central unitaries.

Analyzed relationships between q-deformed classical groups and their extensions.

Provided a framework for further quantum group generalizations.

Abstract

I introduce compact quantum group extensions associated with the $q$-deformations of the classical compact groups $USp(2n)$, $O(n,\mathbb{R})$ and $SO(2n,\mathbb{R})$. Motivated by the relationship between $SU_q(n)$ and $U_q(n)$, I study the problem of constructing compact quantum groups $Z_{q,n}$ extending the standard compact quantum groups $A_{q,n}\in\{ {USp_q(2n), O_q(N), SO_q(2n)}\}$ through an additional central unitary element.