Connections for small vertex models

Abstract

This paper is a first attempt at classifying connections on small vertex models i.e., commuting squares of the form displayed in (1.2) below. More precisely, if we letB(k,n) denote the collection of matricesW for which (1.2) is a commuting square then, we: (i) obtain a simple model form for a representative from each equivalence class inB(2,n), (ii) obtain necessary conditions for two such ‘model connections’ inB(2,n) to be themselves equivalent, (iii) show thatB(2,n) contains a (3n - 6)-parameter family of pairwise inequivalent connections, and (iv) show that the number (3n - 6) is sharp. Finally, we deduce that every graph that can arise as the principal graph of a finite depth subfactor of index 4 actually arises for one arising from a vertex model corresponding toB(2,n) for somen.