On representing some lattices as lattices of intermediate subfactors of finite index

Abstract

We prove that the very simple lattices which consist of a largest, a smallest and 2n pairwise incomparable elements where n is a positive integer can be realized as the lattices of intermediate subfactors of finite index and finite depth. Using the same techniques, we give a necessary and sufficient condition for subfactors coming from Loop groups of type A at generic levels to be maximal.

On maximal subfactors from quantum groups

“…In particular these subfactors are not always maximal. It will be interesting to obtain the result such as Corollary 5.23 of [19] by using the theory of quantum groups at roots of unity.…”

“…When q = 1 is root of unity, one can also construct subfactors as in [16] and [17]. The corresponding subfactors are expected to be related to Jones-Wassermann subfactors, and in §5.2 of [19] the intermediate subfactors are determined by very different method. In particular these subfactors are not always maximal.…”

“…The study of questions which are related to intermediate subfactors has been very active in recent years (cf. [1,5,8,10,12], and [19] for a partial list).…”

Perturbations of intermediate C∗-subalgebras for simple C∗-algebras

“…In the case of subfactor theory of Jones [15], the lattices of intermediate subfactors with their finiteness were studied in Popa [24], Bisch [1], Watatani [27], Teruya and Watatani [25], Longo [20], Khoshkam and Mashhood [19], Grossman and Jones [10], Grossman and Izumi [9] and Xu [28], for example. In these studies, the · 2 -perturbation technique of von Neumann algebras developed by Christensen [6] are essentially used.…”

A quadrilateral in the Asaeda–Haagerup category

“…It is therefore natural to study the lattice of intermediate subfactors of a finite-index subfactor as a quantum analogue of the subgroup lattice of a finite group. The problem of classifying lattices of intermediate subfactors was posed by Watatani [17], and recent progress has been made by Xu [18].…”