Exploration of Finite-Dimensional Kac Algebras and Lattices of Intermediate Subfactors of Irreducible Inclusions

Abstract

Abstract. We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal subalgebras. We reduce the study to KD(n) by proving that the others are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation.Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we describe.

On a subfactor generalization of Wallʼs conjecture

“…We note that recently lattices of intermediate for other types of Kac algebras have been obtained in [6]. Our conjecture can be verified in the examples of [6] where complete lattice of intermediate subfactors are determined.…”

“…We note that recently lattices of intermediate for other types of Kac algebras have been obtained in [6]. Our conjecture can be verified in the examples of [6] where complete lattice of intermediate subfactors are determined. The maximal (or minimal) coideals are very few compared with the dimension of the Kac algebra in these examples of [6].…”