A cluster realization of \(U_q(\mathfrak {sl}_{\mathfrak {n}})\) from quantum character varieties

  • Gus Schrader
  • Alexander ShapiroEmail author


We construct an injective algebra homomorphism of the quantum group \(U_q(\mathfrak {sl}_{n+1})\) into a quantum cluster algebra \(\mathbf {L}_n\) associated to the moduli space of framed \(PGL_{n+1}\)-local systems on a marked punctured disk. We obtain a description of the coproduct of \(U_q(\mathfrak {sl}_{n+1})\) in terms of the corresponding quantum cluster algebra associated to the marked twice punctured disk, and express the action of the R-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of \(U_q(\mathfrak {sl}_{n+1})^{\otimes 2}\) given by conjugation by the R-matrix as an explicit sequence of cluster mutations, and derive a refined factorization of the R-matrix into quantum dilogarithms of cluster monomials.

Mathematics Subject Classification

17B37 13F601 



We would like to thank Vladimir Fock for sharing his insights and expertise, and for his suggestion to study quantum groups via quantization of the moduli spaces of framed local systems considered in his work with Alexander Goncharov. We are grateful to Arkady Berenstein for many inspiring discussions. Our gratitude also goes to Nicolai Reshetikhin for his support and helpful suggestions throughout the course of this work. We thank Linhui Shen for careful reading of the first version of this manuscript, for teaching us a trick used in the proof of Proposition 2, and for providing many excellent remarks, which led in particular to a significant simplification in the exposition of Lemma 7.6. We are grateful to David Hernandez for his hospitality in Paris, where the first version of this paper was completed. Finally, we would like to thank the anonymous referee for valuable comments which helped us improve exposition of the paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of MathematicsUniversity of EdinburghEdinburghUK

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