Séminaires : Séminaire d'Algèbre

Given a split algebraic group G and an oriented surface S with punctures, Fock and Goncharov defined a pair of moduli spaces which parameterise G local systems on S with some extra data at the punctures. These spaces amazingly are Cluster Varieties which endows them with many extra properties, including a well defined set of positive points. They show that these positive points give an algebraic realisation of the Higher Teichmüller space associated to G. 

In this talk I will give an overview of a generalisation of this story to non-split groups G with reduced root systems. Part of this story explains how to think of such groups as a simpler split algebraic group G’ which is defined over a noncommutatve ring, or more correctly a Jordan algebra. We show that the associated moduli spaces have a cluster structure which is a noncommutative deformation of the associated cluster structure for the split group G’. We show that when the Jordan algebras involved are Euclidean i.e. have a positive cone , the group G has a "Theta-positive structure" and there is a well defined set of positive points which parameterise a Higher Teichmüller space associated to G. This gives an algebraic description of the emerging theory of groups with Theta-positive structure. 
 

Based on joint work with Zack Greenberg, Anna Wienhard, and Merik Niemeyer.