A CLUSTER REALIZATION OF Uq(sln) FROM QUANTUM CHARACTER VARIETIES

Gus Schrader, Alexander Shapiro

Research output: Contribution to journalArticlepeer-review

Abstract

We construct an injective algebra homomorphism of the quantum group Uq(sln+1) into a quantum cluster algebra Ln associated to the moduli space of framed PGLn+1-local systems on a marked punctured disk. We obtain a description of the coproduct of Uq(sln+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 Uq(sln+1)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.
Original languageEnglish
Pages (from-to)799-846
Number of pages48
JournalInventiones mathematicae
Early online date19 Jan 2019
DOIs
Publication statusPublished - 1 Jun 2019

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