Abstract

We give a complete description of the generalized Fuss–Catalan algebras: colored generalizations of the Temperley–Lieb algebras, introduced by D. Bisch and V. Jones. For these chains of finite dimensional algebras, we describe a basis in terms of generators, and give a complete description, including the dimensions, of the irreducible representations.

We then consider an arbitrary subfactor containing a chain of intermediate subfactors. The higher relative commutants of a subfactor are an important tool for classifying the subfactor. We show the Fuss–Catalan algebras to be generically contained inside the higher relative commutants of the subfactor. Thus the Fuss–Catalan algebras provide an underlying structure for the higher relative commutants of any subfactor that contains a chain of intermediate subfactors.

Authors