On a subfactor generalization of Wallʼs conjecture

Abstract

In this paper we discuss a conjecture on intermediate subfactors which is a generalization of Wall's conjecture from the theory of finite groups. We explore special cases of this conjecture and present supporting evidence. In particular we prove special cases of this conjecture related to some finite dimensional Kac algebras of Izumi-Kosaki type which include relative version of Wall's conjecture for solvable groups.Published by Elsevier Inc.

On coideal subalgebras of abelian cocentral extensions and a generalization of Wall's conjecture

“…With the above characterization of coideal subalgebras the proof of this Theorem is the same as that of Theorem 3.8 from [4].…”

“…This is compensated by the new characterization of coideal subalgebras given in Theorem 2.1. Also a Hopf algebraic version of the Conjecture 1.1 formulated in [4] follows for cocentral Kac algebras of solvable groups, see Subsection 3.6.…”

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It shown that any coideal subalgebra of a finite dimensional Hopf algebra is a cyclic module over the dual Hopf algebra. Using this we describe all coideal subalgebras of a cocentral abelian extension of Hopf algebras extending some results from [4].

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On maximal subfactors from quantum groups

“…In [20,21] and [7], Conjecture 1.1 is verified for subfactors coming from certain conformal field theories and subfactors which are more closely related to groups and more generally Hopf algebras. The subfactors considered in these papers are finite depth.…”

“…When N ⊂ M comes from group G and its subgroup H, Conjecture 1.1 states that the number of maximal subgroups of G containing H is less than n 1.5 , where n is the number of double cosets of H in G, and it is proved to be true for solvable group G in [7].…”