Representation of the Quantum Plane, Its Quantum Double, and Harmonic Analysis on GLq+(2, ℝ)

Ivan Chi-Ho Ip HKUST

TBD mathscidoc:1909.43013

Selecta Mathematica, 19, (4), 987-1082, 2013
We give complete detail of the description of the GNS representation of the quantum plane A and its dual  as a von Neumann algebra. In particular, we obtain a rather surprising result that the multiplicative unitary W is manageable in this quantum semigroup context. We study the quantum double group construction introduced by Woronowicz, and using Baaj and Vaes' construction of the multiplicative unitary Wm, we give the GNS description of the quantum double D(A) which is equivalent to GLq+(2, ℝ). Furthermore, we study the fundamental corepresentation Tλ, t and its matrix coefficients, and show that it can be expressed by the b-hypergeometric function. We also study the regular corepresentation and representation induced by Wm and prove that the space of L2 functions on the quantum double decomposes into the continuous series representation of Uqq̃(gl(2, ℝ)) with the quantum dilogarithm {pipe}Sb(Q+2iα){pipe}2 as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.
No keywords uploaded!
[ Download ] [ 2019-09-23 15:11:52 uploaded by Ivanip ] [ 449 downloads ] [ 0 comments ]
@inproceedings{ivan2013representation,
  title={Representation of the Quantum Plane, Its Quantum Double, and Harmonic Analysis on GLq+(2, ℝ)},
  author={Ivan Chi-Ho Ip},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190923151152630570504},
  booktitle={Selecta Mathematica},
  volume={19},
  number={4},
  pages={987-1082},
  year={2013},
}
Ivan Chi-Ho Ip. Representation of the Quantum Plane, Its Quantum Double, and Harmonic Analysis on GLq+(2, ℝ). 2013. Vol. 19. In Selecta Mathematica. pp.987-1082. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190923151152630570504.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved