Relative Commutants of Hecke Algebra Subfactors

Abstract

real region and root of unity region display intrinsically different behavior, the most notable example of which is the fact that in the discrete (root of unity) case the relative commutant, which consists of elements of the factor which commute with all of the subfactor, contains only scalar multiples of the identity, while in the continuous (real) case there is a nontrivial relative commutant. This commutant is not there in the algebraic closure of the standard generators, but makes its appearance only in the weak closure. For this reason it is difficult to show the existence of the nontrivial relative commutant [PP86]. Jones' factor forms a representation of the braid group, and the associated Markov trace gives the Jones knot polynomial. Natural generalizations arise by replacing the factor with a quotient of a Hecke algebra. The quotient generates a H,i factor with a Markov trace for the same values of the parameter A, and has a natural subfactor with analogous properties in the discrete and continuous region. Again, no relative commutant is algebraically generated by the standard generators, but it is reasonable to think that there is a relative commutant in the closure in the continuous case (where the closure is taken in the weak topology given by the G.N.S. construction with the Markov trace). That this is true is the principal result of this paper. By imbedding the Hecke algebra quotient inside a much larger algebra, we may express the relative commutant as concrete elements, as well as phrase the question in terms that don't depend on the language of subfactors. We shall think of the Hecke algebra quotient as being a subalgebra of the infinite tensor product of Mr, the algebra of r by r complex matrices, with the Markov trace given by the restriction of a Powers state (the weak closure of the quotient is a von Neumann subalgebra of the weak closure of the tensor product algebra under the G.N.S. construction with this state). The relative commutant consists of the diagonal subalgebra of the first copy of Mr in the tensor product. More generally, the weak closure of the quotient is the fixed point algebra of the modular group