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Three dimensional quantum geometry and deformed symmetry

### Abstract

We study a three dimensional noncommutative space emerging in the context of three dimensional Euclidean quantum gravity. Our starting point is the assumption that the isometry group is deformed to the Drinfeld double . We generalize to the deformed case the construction of as the quotient of its isometry group by . We show that the algebra of functions on becomes the noncommutative algebra of distributions, , endowed with the convolution product. This construction gives the action of on the algebra and allows the determination of plane waves and coordinate functions. In particular, we show the following: (i) plane waves have bounded momenta; (ii) to a given momentum are associated several elements leading to an effective description of in terms of several physical scalar fields on ; (iii) their product leads to a deformed addition rule of momenta consistent with the bound on the spectrum. We generalize to the noncommutative setting the “local” action for a scalar field. Finally, we obtain, using harmonic analysis, another useful description of the algebra as the direct sum of the algebra of matrices. The algebra of matrices inherits the action of : rotations leave the order of the matrices invariant, whereas translations change the order in a way we explicitly determine.

© 2009 American Institute of Physics

Received 20 November 2008
Accepted 14 April 2009
Published online 19 May 2009

Acknowledgments:
We would like to thank Renaud Parentani for discussions in the early stages of the paper. K.N. wants to thank E. Livine and A. Perez for discussions. The work of K.N. was partially supported by the ANR (Grant No. BLAN06-3_139436 LQG-2006). The work of J.M. was partially supported by the EU FP6 Marie Curie Research and Training Network “UniverseNet” (Grant No. MRTN-CT-2006-035863).

Article outline:

I. INTRODUCTION
II. THE QUANTUM DOUBLE : a deformation of the Euclidean group
A. Definition of the quantum double
B. Drinfeld double as a deformation of
1. The classical group algebra
2. Algebra morphisms between and
3. The coalgebra structures: Addition rule of momenta
III. CONVOLUTION ALGEBRA FROM
A. Construction in the classical case
B. Construction in the deformed case
IV. THE NONCOMMUTATIVE ALGEBRA
A. Relation between and
1. Example 1: is a function on
2. Example 2: is an element of
3. Example 3: is a plane wave
4. Example 4: The coordinate functions
B. The -product
C. Invariant measure on and scalar action
V. FUZZY SPACE FORMULATION OF
A. Fourier transform of
B. symmetry in the fuzzy space
1. Action of
2. Action of
VI. RELATION BETWEEN AND
A. The general relation
B. The coordinates on the fuzzy space
C. Examples
1. Constant function
2. Radial function
3. Delta function
D. Maximally localized state
VII. CONCLUSION

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2009-05-19

2016-10-01

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