Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-${N}$
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866918125741539328 |
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| author | Han, Muxin |
| author_facet | Han, Muxin |
| contents | A family of infinite-dimensional irreducible $*$-representations on $\mathcal{H}\simeq L^2(\mathbb{R})\otimes\mathbb{C}^N$ is defined for a quantum-deformed Lorentz algebra $\mathscr{U}_{\bf q}(sl_2)\otimes \mathscr{U}_{\widetilde{\bf {q}}}(sl_2)$, where $\mathbf{q}=\exp[\frac{πi}{N}(1+b^2)]$ and $\tilde{\mathbf{q}}=\exp[\frac{πi}{N}(1+b^{-2})]$ with $N\in\mathbb{Z}_+$ and $|b|=1$. The representations are constructed with the irreducible representation of quantum torus algebra at level-$N$, which is developed from the quantization of $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons theory. We study the Clebsch-Gordan decomposition of the tensor product representation, and we show that it reduces to the same problem as diagonalizing the complex Fenchel-Nielson length operators in quantizing $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons theory on 4-holed sphere. Finally, we explicitly compute the spectral decomposition of the complex Fenchel-Nielson length operators and the corresponding direct-integral representation of the Hilbert space $\mathcal{H}$, which we call the Fenchel-Nielson representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08176 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-${N}$ Han, Muxin High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics Geometric Topology Quantum Algebra A family of infinite-dimensional irreducible $*$-representations on $\mathcal{H}\simeq L^2(\mathbb{R})\otimes\mathbb{C}^N$ is defined for a quantum-deformed Lorentz algebra $\mathscr{U}_{\bf q}(sl_2)\otimes \mathscr{U}_{\widetilde{\bf {q}}}(sl_2)$, where $\mathbf{q}=\exp[\frac{πi}{N}(1+b^2)]$ and $\tilde{\mathbf{q}}=\exp[\frac{πi}{N}(1+b^{-2})]$ with $N\in\mathbb{Z}_+$ and $|b|=1$. The representations are constructed with the irreducible representation of quantum torus algebra at level-$N$, which is developed from the quantization of $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons theory. We study the Clebsch-Gordan decomposition of the tensor product representation, and we show that it reduces to the same problem as diagonalizing the complex Fenchel-Nielson length operators in quantizing $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons theory on 4-holed sphere. Finally, we explicitly compute the spectral decomposition of the complex Fenchel-Nielson length operators and the corresponding direct-integral representation of the Hilbert space $\mathcal{H}$, which we call the Fenchel-Nielson representation. |
| title | Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-${N}$ |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2402.08176 |