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Cluster structures on higher Teichmuller spaces for classical groups. (English) Zbl 1442.13077
Summary: Let $$S$$ be a surface, $$G$$ a simply connected classical group, and $$G'$$ the associated adjoint form of the group. We show that the moduli spaces of framed local systems $$\mathcal{X}_{G',S}$$ and $$\mathcal{A}_{G,S}$$, which were constructed by V. Fock and A. Goncharov [Publ. Math., Inst. Hautes Étud. Sci. 103, 1–211 (2006; Zbl 1099.14025)], have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in that paper, and also allows one to quantize higher Teichmuller space, which was previously only possible when $$G$$ was of type $$A$$.

MSC:
 13F60 Cluster algebras 15A72 Vector and tensor algebra, theory of invariants 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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References:
 [1] Berenstein, A.; Fomin, S.; Zelevinsky, A., Cluster algebras III: upper bounds and double Bruhat cells, Duke Math. J., 126, 1, 1-52, (2005) · Zbl 1135.16013 [2] Cautis, S.; Kamnitzer, J.; Morrison, S., Webs and quantum skew Howe duality, Mathe. Ann., 360, 1-2, 351-390, (2014) · Zbl 1387.17027 [3] Fock, V. V.; Goncharov, A. B., Moduli spaces of local systems and higher Teichmuller theory, Publ. Math. Inst. Hautes Études Sci., 103, 1-212, (2006) · Zbl 1099.14025 [4] Fock, V. V.; Goncharov, A. B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér., 42, 865-929, (2009) · Zbl 1180.53081 [5] Fock, V. V.; Goncharov, A. B., Cluster Ensembles, Quantization and the Dilogarithm II: The Intertwiner, (2007), Manin’s Festschrift: Manin’s Festschrift, Birkhauser [6] Fock, V. V.; Goncharov, A. B., Cluster 𝓧-varieties, amalgamation and Poisson-Lie groups, Algebraic Geometry and Number Theory, 27-68, (2006), Birkhauser: Birkhauser, Boston · Zbl 1162.22014 [7] Fock, V. V.; Goncharov, A. B., The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math., 175, 223-286, (2009) · Zbl 1183.14037 [8] Fomin, S.; Pylyavskyy, P., Webs on surfaces, rings of invariants, and clusters, Proc. Natl. Acad. Sci. USA, 111, 27, 9680-9687, (2014) · Zbl 1355.53012 [9] Fomin, S.; Zelevinsky, A., Double Bruhat cells and total positivity, J. Amer. Math. Soc., 12, 335-380, (1999) · Zbl 0913.22011 [10] Fomin, S.; Zelevinsky, A., Cluster algebras II: finite type classification, Invent. Math., 154, 63-121, (2003) · Zbl 1054.17024 [11] Gaitto, D.; Moore, G.; Neitzke, A., Spectral networks, Ann. Henri Poincaré, 14, 1643-1731, (2013) · Zbl 1288.81132 [12] Goncharov, A. B.; Shen, L., Geometry of canonical bases and mirror symmetry, Invent. Math., 202, 487-633, (2015) · Zbl 1355.14030 [13] Goncharov, A. B.; Shen, L., Donaldson-Thomas transformations of moduli spaces of G-local systems, Invent. Math., 202, 487-633, (2015) [14] Henriques, A. [15] Henriques, A.; Kamnitzer, J., The octaheron recurrence and gl(n) crystals with A. Henriques, Adv. Math., 206, 211-249, (2006) [16] Labourie, F., Anosov flows, surface groups and curves in projective space, Invent. Math., 165, 1, 51-114, (2006) · Zbl 1103.32007 [17] Le., I., Higher laminations and affine buildings, Geom. Topol., 20, 3, 1673-1735 · Zbl 1348.30023 [18] Le, I., An approach to cluster structures on moduli of local systems for general groups, Int. Math. Res. Not. [19] Le, I.; Luo, S. [20] Lusztig, G.; Lehrer, G. I., Total positivity and canonical bases, Algebraic Groups and Lie Groups, 281-295, (1997), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 0890.20034 [21] Lusztig, G., Total positivity in reductive groups, Lie theory and geometry, 531-568, (1994), Birkhauser: Birkhauser, Boston, MA · Zbl 0845.20034 [22] Robbins, D. P.; Rumsey, H., Determinants and alternating-sign matrices, Adv. Math., 62, 169-184, (1986) · Zbl 0611.15008 [23] Speyer, D., Perfect matchings and the octahedron recurrence, J. Algebraic Comb., 25, 3, 309-348, (2007) · Zbl 1119.05092 [24] Teschner, J. [25] Williams, H., Cluster Ensembles and Kac-Moody Groups, Adv. Math., 247, 1-40, (2013) · Zbl 1317.13056 [26] Zickert, C.
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