Cluster X-varieties, amalgamation and Poisson-Lie groups

Details Cluster X-varieties, amalgamation and Poisson-Lie groups Cluster X-varieties, amalgamation and Poisson-Lie groups

2005-08-22

arxiv.org/abs/math/0508408v2

Mathematics

Representation Theory

Minor corrections, some examples added. To appear in the Volume dedicated to V. Drinfeld

Scientific paper

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.

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