Abstract: In recent and upcoming work joint with Anna Wienhard, Zach Greenberg and Merik Niemeyer, we describe a large class of Lie groups as simpler Lie groups “defined over noncommutative rings”, the simplest example expresses the symplectic group SP_2n as as SL_2 over a matrix ring.  We use this description to construct cluster coordinates on moduli spaces of G local systems on surfaces decorated by partial flags of G at the punctures of S. The cluster algebras and varieties which arise this way are noncommutative versions of those coming from Fock and Goncharov cluster coordinates associated to the split Lie group and full flags of this simpler type. I will give an overview of this theory and give an outlook towards some new perspectives on cluster quantization.  

Seminar talk is supported in part by the Mrs. Hepsa Ely Silliman Memorial Fund.