On beta pentagon relations
  The (quantum) pentagon relation underlies the existing constructions of three dimensional quantum topology in the combinatorial framework of triangulations. Following the recent works \cite{KashaevLuoVartanov2012,AndersenKashaev2013}, we discuss a special type of integral pentagon relations and their relationships with the Faddeev type operator pentagon relations. 
Rinat Kashaev
math-ph math.MP math.QA