Fock--Goncharov coordinates for semisimple Lie groups

TL;DR

This paper extends Fock--Goncharov's cluster coordinate framework to all semisimple Lie groups, including the exceptional types, providing a comprehensive construction for surface and 3-manifold representations.

Contribution

It offers a complete, unified construction of Fock--Goncharov coordinates for all semisimple Lie groups, filling gaps for exceptional types using Fomin-Zelevinsky's methods.

Findings

Unified coordinate framework for all semisimple Lie groups

Extension to exceptional Lie groups $F_4$, $E_6$, $E_7$, $E_8$

Application to surface and 3-manifold representations

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 $F_4$, $E_6$, $E_7$, and $E_8$.