Cluster Realization of Uq(g) and Factorizations of The Universal R-matrix

Ivan Chi-Ho Ip HKUST

TBD mathscidoc:1909.43023

Selecta Mathematica, New Series, 24, (5), 2018
For each simple Lie algebra g, we construct an algebra embedding of the quantum group Uq(g) into certain quantum torus algebra Dg via the positive representations of split real quantum group. The quivers corresponding to Dg is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, and show that conjugation by the R-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.
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@inproceedings{ivan2018cluster,
  title={Cluster Realization of Uq(g) and Factorizations of The Universal R-matrix},
  author={Ivan Chi-Ho Ip},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190923160204522440514},
  booktitle={Selecta Mathematica, New Series},
  volume={24},
  number={5},
  year={2018},
}
Ivan Chi-Ho Ip. Cluster Realization of Uq(g) and Factorizations of The Universal R-matrix. 2018. Vol. 24. In Selecta Mathematica, New Series. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190923160204522440514.
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