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

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Abstract

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.

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Authors and Affiliations

  1. Center for the Promotion of Interdisciplinary Education and Research/Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, Japan

    Ivan C. H. Ip

  2. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

    Ivan C. H. Ip

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  1. Ivan C. H. Ip
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Ip, I.C.H. Cluster realization of \(\mathcal {U}_q(\mathfrak {g})\) and factorizations of the universal R-matrix. Sel. Math. New Ser. 24, 4461–4553 (2018). https://doi.org/10.1007/s00029-018-0432-0

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