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

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 ${\delta }_{\mathcal{H}}{G}_q$ compatible with the Hopf algebra structure. It is shown that ${\delta }_{\mathcal{H}}{G}_q$ exists only for the quantum group $U_q(N)$ and that the quantum group $S{U}_q(N)$ as a quantum gauge group is not allowed. The representations of the algebra ${\delta }_{\mathcal{H}}{G}_q$ 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.

14 pages (Changes in Introduction+changes in References+change in title)

References in corpus (1)

• Quantum Group Gauge Theories and Covariant Quantum Algebras

Cited by in corpus (1)

• Poisson-Lie structures on the external algebra of $SL(2)$ and their quantization