Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions

TL;DR

This paper investigates finite dimensional Kac algebras, especially the KD(n) family, and explores their associated lattices of intermediate subfactors, automorphisms, and dualities, supported by computational methods.

Contribution

It reduces the study of four Kac algebra families to KD(n), develops tools for analyzing coideal subalgebras, and explores lattices of intermediate subfactors with computational techniques.

Findings

Classified automorphism groups of Kac algebra families

Derived lattices of intermediate subfactors and principal graphs

Extended Galois correspondence for depth 2 inclusions

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.