Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-N
Han M (2025)
Publication Type: Journal article
Publication year: 2025
Journal
Book Volume: 66
Article Number: 073508
Journal Issue: 7
DOI: 10.1063/5.0207625
Abstract
A family of infinite-dimensional irreducible ∗-representations on H ≃ L 2 ( R ) ⊗ C N is defined for a quantum-deformed Lorentz algebra U q ( s l 2 ) ⊗ U q ̃ ( s l 2 ) , where q = exp [ π i N ( 1 + b 2 ) ] and q ̃ = exp [ π i N ( 1 + b − 2 ) ] with N ∈ 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 S L ( 2 , 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 S L ( 2 , 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 H , which we call the Fenchel-Nielson representation.
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APA:
Han, M. (2025). Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-N. Journal of Mathematical Physics, 66(7). https://doi.org/10.1063/5.0207625
MLA:
Han, Muxin. "Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-N." Journal of Mathematical Physics 66.7 (2025).
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