Applications of the strong free product property to symplectic topology and multiplicative quiver varieties

Seminar: 
Clusters and Geometry
Event time: 
Friday, November 6, 2020 - 1:00pm
Speaker: 
Daniel Kaplan
Speaker affiliation: 
University of Birmingham
Event description: 

In 2006, Crawley-Boevey and Shaw defined the multiplicative preprojective algebra (MPA) to study certain character varieties. More recently, MPAs appeared in work of Etgü--Lekili in the study of Fukaya categories of 4-manifolds. Nice properties of the (additive) preprojective algebra are expected to hold for MPAs, but most proof techniques are not available. In joint work with Travis Schedler, we define a new property, called the strong free product property, that allows us to prove MPAs are 2-Calabi--Yau algebras and quasi-isomorphic to their dg counterparts, for quivers containing a cycle. Moreover, using a result of Bocklandt--Galluzzi--Vaccarino, we prove the formal local structure of multiplicative quiver varieties is isomorphic to that of a (usual) quiver variety. In this talk, I'll survey these ideas and illustrate them in small examples.