[1612.05641] Cluster Realization Of $U_q(\mathfrak{g})$ And ... - ArXiv

Mathematics > Quantum Algebra arXiv:1612.05641 (math) [Submitted on 16 Dec 2016 (v1), last revised 16 Feb 2017 (this version, v3)] Title:Cluster Realization of $U_q(\mathfrak{g})$ and Factorization of the Universal $R$-Matrix Authors:Ivan Chi-Ho Ip View a PDF of the paper titled Cluster Realization of $U_q(\mathfrak{g})$ and Factorization of the Universal $R$-Matrix, by Ivan Chi-Ho Ip View PDF

Abstract:For each simple Lie algebra $\mathfrak{g}$, we construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into certain quantum torus algebra $D_\mathfrak{g}$ via the positive representations of split real quantum group. The quivers corresponding to $D_\mathfrak{g}$ is obtained from amalgamation of two basic quivers, where each of them is mutation equivalent to the cluster structure of the moduli space of framed $G$-local system on a disk with 3 marked points when $G$ is of classical type. We derive a factorization of the universal $R$-matrix into quantum dilogarithms of cluster variables, 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.

Comments:	Extended introduction and added more references. Added Remarks 4.16, 6.2 and 8.6. Fixed a misprint about co-product
Subjects:	Quantum Algebra (math.QA); Representation Theory (math.RT)
MSC classes:	17B37, 13F60
Cite as:	arXiv:1612.05641 [math.QA]
(or arXiv:1612.05641v3 [math.QA] for this version)
https://doi.org/10.48550/arXiv.1612.05641 Focus to learn more arXiv-issued DOI via DataCite

Submission history

From: Ivan Chi-Ho Ip [view email] [v1] Fri, 16 Dec 2016 21:00:01 UTC (57 KB) [v2] Fri, 3 Feb 2017 11:04:23 UTC (59 KB) [v3] Thu, 16 Feb 2017 10:01:28 UTC (59 KB) Full-text links:

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