The Euclidean Hopf algebra $U_q(e^N)$ and its fundamental Hilbert space representations

Abstract

We construct the Euclidean Hopf algebra $U_q(e^N)$ dual of $Fun(\rn_q^N\lcross SO_{q^{-1}}(N))$ by realizing it as a subalgebra of the differential algebra $\DFR$ on the quantum Euclidean space $\rn_q^N$; in fact, we extend our previous realization \cite{fio4} of $U_{q^{-1}}(so(N))$ within $\DFR$ through the introduction of q-derivatives as generators of q-translations. The fundamental Hilbert space representations of $U_q(e^N)$ turn out to be of highest weight type and rather simple `` lattice-regularized '' versions of the classical ones. The vectors of a basis of the singlet (i.e. zero-spin) irrep can be realized as normalizable functions on $\rn_q^N$, going to distributions in the limit $q\to 1$.

67 pages, 1 figures. Revised version: Format changed, typos amended, some citations added

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Cited by in corpus (15)

• Identical Particles and Quantum Symmetries
• Realization of $U_q(so(N))$ within the differntial algebra on ${\mathbf{R}}_q^N$
• Structure of the Three-dimensional Quantum Euclidean Space
• On the Decoupling of the Homogeneous and Inhomogeneous Parts in Inhomogeneous Quantum Groups
• Fredholm Modules for Quantum Euclidean Spheres
• On the deformability of Heisenberg algebras
• The q-Euclidean algebra $U_q(e^N)$ and the corresponding q-Euclidean lattice
• Statistics and Quantum Group Symmetries
• On the hermiticity of q-differential operators and forms on the quantum Euclidean spaces R_q^N
• Many Q-Particles from One: A New Approach to $\ast$-Hopf Algebras?
• Noncommutative Spheres and Instantons
• Twisted Configurations over Quantum Euclidean Spheres
• Geometrical Issues for the 3-dim Quantum Euclidean Space
• Hilbert space representations of cross product algebras II
• New approach to Hermitian q-differential operators on R_q^N