Realization of $U_q(so(N))$ within the differntial algebra on ${\mathbf{R}}_q^N$

Abstract

We realize the Hopf algebra $U_{q^{-1}}(so(N))$ as an algebra of differential operators on the quantum Euclidean space ${\mathbf{R}}_q^N$. The generators are suitable q-deformed analogs of the angular momentum components on ordinary ${\mathbf{R}}^N$. The algebra $Fun({\mathbf{R}}_q^N)$ of functions on ${\mathbf{R}}_q^N$ splits into a direct sum of irreducible vector representations of $U_{q^{-1}}(so(N))$; the latter are explicitly constructed as highest weight representations.

26 pages, 1 figure

References in corpus (3)

• Reality in the Differential Calculus on q-euclidean Spaces
• The $S{O}_q(N,\mathbf{R})$-Symmetric Harmonic Oscillator on the Quantum Euclidean Space ${\mathbf{R}}_q^N$ and its Hilbert Space Structure
• The Euclidean Hopf algebra $U_q(e^N)$ and its fundamental Hilbert space representations

Cited by in corpus (12)

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• Unbraiding the braided tensor product
• On the Decoupling of the Homogeneous and Inhomogeneous Parts in Inhomogeneous Quantum Groups
• The q-Euclidean algebra $U_q(e^N)$ and the corresponding q-Euclidean lattice
• On the hermiticity of q-differential operators and forms on the quantum Euclidean spaces R_q^N
• Representations of $U_h(su(N))$ derived from quantum flag manifolds
• The Fourier Transform on Quantum Euclidean Space
• Decoupling Braided Tensor Factors
• New approach to Hermitian q-differential operators on R_q^N
• Realization of $U_q({\mathfrak{sp}}_{2n})$ within the Differential Algebra on Quantum Symplectic Space