Quantum deformation of Lorentz group |
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Authors: | P Podleś 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
L(2,) ofSL(2,) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition coincides withS
U(2), whereas the solvable part is identified as a Pontryagin dual ofS
U(2). It shows thatS
L(2,) is the result of the dual version of Drinfeld's double group construction applied toS
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
L(2,) is the same as that ofSL(2,) (Clebsh-Gordon coefficients are however modified). The corresponding tame bicovariant bimodules onS
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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