Ore's theorem for cyclic subfactor planar algebras and applications

Abstract

This paper introduces the cyclic subfactors, generalizing the cyclic groups as the subfactors generalize the groups, and generalizing the natural numbers as the maximal subfactors generalize the prime numbers. On one hand, a theorem of O. Ore states that a finite group is cyclic if and only if its subgroups lattice is distributive, and on the other hand, every subgroup of a cyclic group is normal. Then, a subfactor planar algebra is called cyclic if all the biprojections are normal and form a distributive lattice. The main result shows in what sense a cyclic subfactor is singly generated, by generalizing one side of Ore's theorem as follows: if a subfactor planar algebra is cyclic then it is weakly cyclic (or w-cyclic), i.e. there is a minimal 2-box projection generating the identity biprojection. Some extensions of this result are discussed, and some applications of it are given in subfactors, quantum groups and finite group theories, for example, a non-trivial upper bound for the minimal number of irreducible complex representations generating the left regular representation.