Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions
arXiv:0812.3044 · doi:10.1142/S0219498811005099
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 further describe.
arXiv commentv1: 84 pages, 13 figures, submitted. v2: 94 pages, 15 figures, added connections with Masuoka's families KA and KB, description of K3 in KD(n), lattices for KD(8) and KD(15). v3: 93 pages, 15 figures, proven lattice for KD(6), misc improvements, accepted for publication in Journal of Algebra and Its Applications