Event № 157
Cluster algebras are commutative rings with a distinguished set of generators that are grouped into overlapping finite sets of the same cardinality. Among many other examples, cluster algebras appear in coordinate rings of various algebraic varieties.
Using the notion of compatibility between Poisson brackets and cluster algebras in the coordinate rings of simple complex Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang Baxter equation. For a simple complex Lie group G and a Belavin-Drinfeld class, one can define a corresponding Poisson bracket on the ring of regular functions on G. For certain types of classes in SLn, a compatible cluster structure can be constructed. Cluster algebras will be defined and explained, and a description of compatible structures and their properties will be given.