A pentagon of identities, graded tensor products and the Kirillov-Reshetikhin conjecture

This paper provides a brief review of the relations between the Feigin-Loktev conjecture on the dimension of graded tensor products of \g[t]-modules, the Kirillov-Reshetikhin conjecture, the combinatorial ``M=N" conjecture, their proofs for all simple Lie algebras, and a pentagon of identities which results from the proof.Comment: 21 page

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.Comment: 11 pages, Theorem 2 generalized to arbitrary LCA groups and renumbered to Theorem