Noncommutative coordinates for symplectic representations


We introduce coordinates on the space of Lagrangian decorated and framed representations of the fundamental group of a surface with punctures into the symplectic group Sp(2n,R). These coordinates provide a non-commutative generalization of the parametrizations of the spaces of representations into \mathrmSL(2,R) given by Thurston, Penner, and Fock-Goncharov. With these coordinates, the space of framed symplectic representations provides a geometric realization of the non-commutative cluster algebras introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of decorated maximal representations. We use this to determine the homotopy type of the space of decorated maximal representations, and its homeomorphism type when n=2.


Research Groups: Groups and Geometry

Publication type: Misc

Journal: arXiv,math.DG,1911.08014

Citation: arXiv,math.DG,1911.08014

Date Published: 2019


Registered Mode: imported from a bibtex file

Authors: Daniele Alessandrini, Olivier Guichard, Evgenii Rogozinnikov, Anna Wienhard

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Created: 30th Jan 2020 at 09:49

Last updated: 5th Mar 2024 at 21:24

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