Harmonic Analysis on the Quantum Lorentz Group

Buffenoir, Eric; Roche, Ph.

doi:10.1007/s002200050736

Cited by 44 publications

(180 citation statements)

References 35 publications

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“…An introduction to the notion of complexification and realification in the Hopf algebra context can be found in the chapter 2 of [13].…”

Section: Lemmamentioning

confidence: 99%

“…We can define a Fock-Rosly structure on them, the Poisson bracket is the same as (13,14,15) where the classical r matrix of su (2) has been replaced by the r matrix of sl(2, C) R : r sl(2,C) R = r ll 12 − r rr 21 . We refer the reader to the article ( [12,13]) for a thorough study of the quantum group U q (sl(2, C) R ), see also the appendix (A.1) where basic definitions as well as fundamental results on harmonic analysis are described.…”

Section: For Any Representationmentioning

confidence: 99%

“…Thanks to the factorisation theorem, i.e. (2)) as a Hopf algebra [12,13], R is expressed in term of U q (sl(2)) R-matrices as…”

Section: For Any Representationmentioning

confidence: 99%

“…Given α ∈ S, we can define the character ω 1,0 (α) ∈ (F un cc (SL q (2, C))) * of this representation by ( [12]):…”

Section: Iii22 the Moduli Algebra Of A Genus-n Surfacementioning

confidence: 99%

“…We will give a summary of the harmonic analysis on SL q (2, C) R , for a complete treatment see [12,13].…”

Section: A1 Representations and Harmonic Analysismentioning

confidence: 99%

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Hamiltonian quantization of Chern Simons theory with SL (2, ) group

Buffenoir¹,

Noui²,

Roche³

2002

Class. Quantum Grav.

160

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We analyze the hamiltonian quantization of Chern-Simons theory associated to the real group SL(2, C) R , universal covering of the Lorentz group SO(3, 1). The algebra of observables is generated by finite dimensional spin networks drawn on a punctured topological surface. Our main result is a construction of a unitary representation of this algebra. For this purpose we use the formalism of combinatorial quantization of Chern-Simons theory, i.e we quantize the algebra of polynomial functions on the space of flat SL(2, C) R −connections on a topological surface Σ with punctures. This algebra, the so called moduli algebra, is constructed along the lines of Fock-Rosly, Alekseev-Grosse-Schomerus, Buffenoir-Roche using only finite dimensional representations of U q (sl(2, C) R ). It is shown that this algebra admits a unitary representation acting on an Hilbert space which consists in wave packets of spin-networks associated to principal unitary representations of U q (sl(2, C) R ). The representation of the moduli algebra is constructed using only Clebsch-Gordan decomposition of a tensor product of a finite dimensional representation with a principal unitary representation of U q (sl(2, C) R ). The proof of unitarity of this representation is non trivial and is a consequence of properties of U q (sl(2, C) R ) intertwiners which are studied in depth. We analyze the relationship between the insertion of a puncture colored with a principal representation and the presence of a world-line of a massive spinning particle in de Sitter space.

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“…An introduction to the notion of complexification and realification in the Hopf algebra context can be found in the chapter 2 of [13].…”

Section: Lemmamentioning

confidence: 99%

Section: For Any Representationmentioning

confidence: 99%

“…Thanks to the factorisation theorem, i.e. (2)) as a Hopf algebra [12,13], R is expressed in term of U q (sl(2)) R-matrices as…”

Section: For Any Representationmentioning

confidence: 99%

“…Given α ∈ S, we can define the character ω 1,0 (α) ∈ (F un cc (SL q (2, C))) * of this representation by ( [12]):…”

Section: Iii22 the Moduli Algebra Of A Genus-n Surfacementioning

confidence: 99%

“…We will give a summary of the harmonic analysis on SL q (2, C) R , for a complete treatment see [12,13].…”

Section: A1 Representations and Harmonic Analysismentioning

confidence: 99%

See 3 more Smart Citations

Hamiltonian quantization of Chern Simons theory with SL (2, ) group

Buffenoir¹,

Noui²,

Roche³

2002

Class. Quantum Grav.

160

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From general relativity to quantum gravity

Ashtekar¹,

Reuter

Rovelli

Mathematical Problems in Theoretical Physics

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In general relativity (GR), spacetime geometry is no longer just a background arena but a physical and dynamical entity with its own degrees of freedom. We present an overview of approaches to quantum gravity in which this central feature of GR is at the forefront. However, the short distance dynamics in the quantum theory are quite different from those of GR and classical spacetimes and gravitons emerge only in a suitable limit. Our emphasis is on communicating the key strategies, the main results and open issues. In the spirit of this volume, we focus on a few avenues that have led to the most significant advances over the past 2-3 decades. 1

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Combinatorial quantization of Euclidean gravity in three dimensions

Schroers

2001

Quantization of Singular Symplectic Quotients

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In the Chern-Simons formulation of Einstein gravity in 2+1 dimensions the phase space of gravity is the moduli space of flat G-connections, where G is a typically non-compact Lie group which depends on the signature of space-time and the cosmological constant. For Euclidean signature and vanishing cosmological constant, G is the three-dimensional Euclidean group. For this case the Poisson structure of the moduli space is given explicitly in terms of a classical r-matrix. It is shown that the quantum R-matrix of the quantum double D(SU (2)) provides a quantisation of that Poisson structure.MSC 17B37, 81R50, 81S10, 83C45

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Harmonic Analysis on the Quantum Lorentz Group

Cited by 44 publications

References 35 publications

Hamiltonian quantization of Chern Simons theory with SL (2, ) group

Hamiltonian quantization of Chern Simons theory with SL (2, ) group

From general relativity to quantum gravity

Combinatorial quantization of Euclidean gravity in three dimensions

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