Cluster X-varieties, amalgamation and Poisson-Lie groups

TL;DR

This paper constructs cluster X-varieties associated with split semisimple Lie groups, explores their Poisson structures, and introduces an amalgamation operation, linking them to double Bruhat cells and moduli spaces of local systems.

Contribution

It defines a new class of cluster X-varieties for Lie groups, introduces an amalgamation operation, and connects these structures to Poisson-Lie groups and double Bruhat cells.

Findings

Cluster X-varieties are Poisson varieties related to Lie groups.

Amalgamation of elementary cluster varieties is introduced.

Connections to double Bruhat cells and moduli spaces are established.

Abstract

Starting from a split semisimple real Lie group G with trivial center, we define a family of varieties with additional structures. We describe them as the cluster X-varieties, as defined in math.AG/0311245. In particular they are Poisson varieties. We define Poisson maps of them to the group G with the standard Poisson-Lie structure. We introduce an operation of amalgamation of cluster varieties. Our varieties are amalgamations of elementary ones, assigned to positive simple roots of the root system of G. Some of them are very closely related to the double Bruhat cells. This paper is a building block in a description of the cluster structure of the moduli spaces of local systems on surfaces studied in math.AG/0311149.