Selecta Mathematica

, Volume 24, Issue 5, pp 4461–4553 | Cite as

Cluster realization of \(\mathcal {U}_q(\mathfrak {g})\) and factorizations of the universal R-matrix

  • Ivan C. H. IpEmail author


For each simple Lie algebra \(\mathfrak {g}\), we construct an algebra embedding of the quantum group \(\mathcal {U}_q(\mathfrak {g})\) into certain quantum torus algebra \(\mathcal {D}_\mathfrak {g}\) via the positive representations of split real quantum group. The quivers corresponding to \(\mathcal {D}_\mathfrak {g}\) 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.


Quantum groups Positive representations Cluster algebra R-matrix 

Mathematics Subject Classification

Primary 17B37 13F60 


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Center for the Promotion of Interdisciplinary Education and Research/Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan
  2. 2.Department of MathematicsHong Kong University of Science and TechnologyClear Water BayHong Kong

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