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Positive Representations of Split Real Quantum Groups and Future Perspectives

Igor Frenkel Yale University Ivan Chi-Ho Ip HKUST

TBD mathscidoc:1909.43024

International Mathematics Research Notices, 2014, 2014
We construct a special principal series representation for the modular double Uqq∼(gR) of type Ar representing the generators by positive essentially self-adjoint operators satisfying the transcendental relations that also relate q and. We use the cluster variables parameterization of the positive unipotent matrices to derive the formulas in the classical case. Then we quantize them after applying the Mellin transform. Our construction is inspired by the previous results for gR =sl(2,R) and can be generalized to all other types of simple split real Lie algebra. We conjecture that our positive representations are closed under the tensor product and we discuss the future perspectives of the new representation theory following the parallel with the established developments of the finite-dimensional representation theory of quantum groups.
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@inproceedings{igor2014positive,
  title={Positive Representations of Split Real Quantum Groups and Future Perspectives},
  author={Igor Frenkel, and Ivan Chi-Ho Ip},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190923160744969137515},
  booktitle={International Mathematics Research Notices},
  volume={2014},
  year={2014},
}
Igor Frenkel, and Ivan Chi-Ho Ip. Positive Representations of Split Real Quantum Groups and Future Perspectives. 2014. Vol. 2014. In International Mathematics Research Notices. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190923160744969137515.
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