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CLUSTER STRUCTURES ON HIGHER TEICHMULLER SPACES FOR CLASSICAL GROUPS

Part of: Basic linear algebra Arithmetic rings and other special rings

Published online by Cambridge University Press:  08 May 2019

IAN LE
IAN LE*
Affiliation:
Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada; ile@perimeterinstitute.ca

Abstract

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Let $S$ be a surface, $G$ a simply connected classical group, and $G^{\prime }$ the associated adjoint form of the group. We show that the moduli spaces of framed local systems ${\mathcal{X}}_{G^{\prime },S}$ and ${\mathcal{A}}_{G,S}$, which were constructed by Fock and Goncharov [‘Moduli spaces of local systems and higher Teichmuller theory’, Publ. Math. Inst. Hautes Études Sci.103 (2006), 1–212], 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 classification

Primary: 13F60: Cluster algebras
Secondary: 15A72: Vector and tensor algebra, theory of invariants
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e13
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2019

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