A Categorification of Hall Algebras

摘要: In recent years, there has been great interest in the study of categorification, specifically as it applies to theory quantum groups. this thesis, we would like provide a new approach problem by looking at Hall algebras. It is know, due Ringel, that algebra isomorphic certain group. our goal describe categorification algebras way doing so for their related To do this, will take following steps. First, perspective on structure This view solves, unique way, classic multiplication and comultiplication not being compatible. Our solution switch different underlying category Vect^K vector spaces graded group K called Grothendieck We equip with nontrivial braiding which depends K-grading. With given antipode, find does become Hopf object Vect^K. Second, process, call `groupoidification', replaces groupoids linear operators `spans' groupoids. use process construct braided monoidal bicategory categorifies via groupoidification program. Specifically, be replaced `over' fixed groupoid K. The come from an interesting EXT(M,N) behave Euler characteristic finish description plan to, future work, apply same concept maps algebra, eventually give us 2-algebra bicategory.