Quantum deformation of lorentz group |
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Authors: | P Podle? S L Woronowicz |
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Institution: | (1) Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoza 74, PL-00-682 Warszawa, Poland |
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Abstract: | A one parameter quantum deformationS
μ
L(2,ℂ) ofSL(2,ℂ) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition
coincides withS
μ
U(2), whereas the solvable part is identified as a Pontryagin dual ofS
μ
U(2). It shows thatS
μ
L(2,ℂ) is the result of the dual version of Drinfeld's double group construction applied toS
μ
U(2). The same construction applied to any compact quantum groupG
c
is discussed in detail. In particular the explicit formulae for the Haar measures on the Pontryagin dualG
d
ofG
c
and on the double groupG are given. We show that there exists remarkable 1-1 correspondence between representations ofG and bicovariant bimodules (“tensor bundles”) overG
c
. The theory of smooth representations ofS
μ
L(2,ℂ) is the same as that ofSL(2,ℂ) (Clebsh-Gordon coefficients are however modified). The corresponding “tame” bicovariant bimodules onS
μ
U(2) are classified. An application to 4D
+ differential calculus is presented. The nonsmooth case is also discussed. |
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Keywords: | |
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