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.
