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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 \(A\).

13F60 Cluster algebras
15A72 Vector and tensor algebra, theory of invariants
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
Full Text: DOI
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