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

Gus Schrader; Alexander Shapiro

doi:10.1007/s00222-019-00857-6

A CLUSTER REALIZATION OF U_q(sl_n) FROM QUANTUM CHARACTER VARIETIES

Gus Schrader, Alexander Shapiro

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We construct an injective algebra homomorphism of the quantum group U_q(sl_n+1) into a quantum cluster algebra 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(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(sl_n+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 language	English
Pages (from-to)	799-846
Number of pages	48
Journal	Inventiones mathematicae
Early online date	19 Jan 2019
DOIs	https://doi.org/10.1007/s00222-019-00857-6
Publication status	Published - 1 Jun 2019

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https://arxiv.org/abs/1607.00271

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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.",

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N2 - 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.

AB - 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.

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JO - Inventiones mathematicae

JF - Inventiones mathematicae

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

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A CLUSTER REALIZATION OF U_q(sl_n) FROM QUANTUM CHARACTER VARIETIES