Representation theory of Chern-Simons observables

Alekseev, Anton; Schomerus, Volker

doi:10.1215/s0012-7094-96-08519-1

Cited by 64 publications

(175 citation statements)

References 42 publications

(45 reference statements)

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“…Fun cc (SL q (2; C ) R ) = Pol(SU 0 q (2) Fun c (AN q (2)) is a multiplier Hopf algebra, let us de ne on the vector spaceD = Pol(SU q (2)) Pol(SU q (2)); where Pol(SU q (2)) is the restricted dual of Pol(SU q (2)) and is the algebraic tensor product, a structure of multiplier Hopf algebra We have already de ned R . and L .…”

Section: Charactersmentioning

confidence: 99%

“…Of course we have to show that this de nition is a norm. This is clearly the case because Pol(SU q (2)) is generated by the coe cients of 1 2 g = a b qb ? a ?…”

Section: Introductionmentioning

confidence: 99%

“…algebra approach developed by Woronowicz and his collaborators, and the R matrix approach developed by the Russian school. We have tried to think of U q (sl(2; C ) R ) as being the double of U q (su (2)) and to forget about commutation relations; this has the advantage that all the results can be recast in terms of generalized 6j symbols and that the proofs are mainly graphical. We not only gain in clarity but obtain results like the description of characters and the Plancherel theorem, which would have been hard to obtain outside this formulation.…”

Section: Introductionmentioning

confidence: 99%

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Harmonic Analysis on the Quantum Lorentz Group

Buffenoir¹,

Roche²

1999

Communications in Mathematical Physics

178

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This work begins with a review of complexi cation and reali cation of Hopf algebras. We emphasize the notion of multiplier Hopf algebras for the description of di erent classes of functions (compact supported, bounded, unbounded) on complex quantum groups and the construction of the associated left and right Haar measure. Using a continuation of 6j symbols of SUq (2) with complex spins, we give a new description of the unitary representations of SLq (2; C ) R and nd explicit expressions for the characters of SLq (2; C ) R . The major theorem of this article is the Plancherel theorem for the Quantum Lorentz Group.

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Section: Charactersmentioning

confidence: 99%

“…Of course we have to show that this de nition is a norm. This is clearly the case because Pol(SU q (2)) is generated by the coe cients of 1 2 g = a b qb ? a ?…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Harmonic Analysis on the Quantum Lorentz Group

Buffenoir¹,

Roche²

1999

Communications in Mathematical Physics

178

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“…As before we shall consider only the restricted set Obs consisting of (the quantum versions of) the classical integrals of motion trA 2 j and A ∞ . As we will see the quantization of the Schlesinger equations in terms of A j leads to the KZ equations; then the quantum monodromies turn out to coincide with the monodromies of the KZ equations up to similarity transformations and to carry representations of certain quantum group [49,50].…”

Section: Hilbert Space Physical States and Quantum Observablesmentioning

confidence: 99%

Isomonodromic quantization of dimensionally reduced gravity

Korotkin¹,

Nicolai²

1996

Nuclear Physics B

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Abstract. We present a detailed account of the isomonodromic quantization of dimensionally reduced Einstein gravity with two commuting Killing vectors. This theory constitutes an integrable "midi-superspace" version of quantum gravity with infinitely many interacting physical degrees of freedom. The canonical treatment is based on the complete separation of variables in the isomonodromic sectors of the model. The Wheeler-DeWitt and diffeomorphism constraints are thereby reduced to the Knizhnik-Zamolodchikov equations for SL(2, R). The physical states are shown to live in a well defined Hilbert space and are manifestly invariant under the full diffeomorphism group. An infinite set of independent observablesà la Dirac exists both at the classical and the quantum level. Using the discrete unitary representations of SL(2, R), we construct explicit quantum states. However, satisfying the additional constraints associated with the coset space SL(2, R)/SO(2) requires solutions based on the principal series representations, which are not yet known. We briefly discuss the possible implications of our results for string theory.

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“…• we can modify the construction of the representation of the moduli algebra in the compact case [15] to apply it to the quantum Lorentz group case. We will therefore obtain non trivial result in 2+1 quantum gravity with positive cosmological constant.…”

Section: Harmonic Analysis Onmentioning

confidence: 99%

Quantum groups and quantum moduli spaces of flat connections

Roché¹

2000

Proceedings of Non-Perturbative Quantum Effects 2000 — PoS(tmr2000)

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Abstract:We give an introduction to the theory of dynamical quantum groups and precise its relation to the harmonic analysis on non-compact quantum groups through the example of the quantum Lorentz-Group.Keywords: Dynamical quantum groups, Non-compact quantum groups, Racah coefficients.Representation of non compact quantum groups is of central importance in different theoretical physics systems. We can at least give three examples of this fact:1. Chern-Simons theory with non compact group G. This is particularly important in view of its applications to quantum gravity in 2+1 dimensions where the group G is equal to SO(3, 1) (Λ > 0), ISO(2, 1) (Λ = 0) or SL(2, R) × SL(2, R), (Λ < 0) depending on the sign of the cosmological constant Λ.2. Discretization of Lorentzian gravity in the spirit of Ponzano-Regge.3. Liouville theory in the weak and probably strong coupling regime.In these three cases, there is an associated "non compact" quantum group, which is a star Hopf algebra, with a category of unitary representations. The computation of physical quantities in these three systems amounts to compute explicitely, or to have a good understanding of the

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Representation theory of Chern-Simons observables

Cited by 64 publications

References 42 publications

Harmonic Analysis on the Quantum Lorentz Group

Harmonic Analysis on the Quantum Lorentz Group

Isomonodromic quantization of dimensionally reduced gravity

Quantum groups and quantum moduli spaces of flat connections

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