Uniqueness of $U_q(N)$ as a quantum gauge group and representations of its differential algebra

TL;DR

This paper demonstrates that the quantum group $U_q(N)$ uniquely admits a compatible differential algebra for gauge theory construction, excluding $SU_q(N)$, and constructs its representations to describe infinitesimal gauge transformations.

Contribution

It proves the uniqueness of $U_q(N)$ as a quantum gauge group with a compatible differential algebra and constructs its representations for gauge transformation analysis.

Findings

$ abla_{ ext{diff}} G_q$ exists only for $U_q(N)$

$SU_q(N)$ cannot serve as a quantum gauge group

Constructed representations describe infinitesimal gauge transformations

Abstract

To construct a quantum group gauge theory one needs an algebra which is invariant under gauge transformations. The existence of this invariant algebra is closely related with the existence of a differential algebra $\delta _{{\cal H}} G_{q}$ compatible with the Hopf algebra structure. It is shown that $\delta _{{\cal H}} G_{q}$ exists only for the quantum group $U_{q}(N)$ and that the quantum group $SU_q(N)$ as a quantum gauge group is not allowed. The representations of the algebra $\delta _{{\cal H}} G_{q}$ are con- structed. The operators corresponding to the differentials are realized via derivations on the space of all irreducible *-representations of $U_q(2)$. With the help of this construction infinitesimal gauge transformations in two-dimensional classical space-time are described.