Fock--Goncharov coordinates for semisimple Lie groups

2021/04/11 by S. Gilles, Gilles, S.
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2104.04941

openalex publication_date 2021/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fock and Goncharov introduced cluster ensembles, providing a framework for coordinates on varieties of surface representations into Lie groups, as well as a complete construction for groups of type A_n. Later, Zickert, Le, and Ip described, using differing methods, how to apply this framework for other Lie group types. Zickert also showed that this framework applies to triangulated 3-manifolds. We present a complete, general construction, based on work of Fomin and Zelevinsky. In particular, we complete the picture for the remaining cases: Lie groups of types F₄, E₆, E₇, and E₈.

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