Kac algebras arising from composition of subfactors: General theory and classification

We deal with a map a from a finite group G into the automorphism group Aut(L) of a factor L satisfying (i) G = N x H is a semi-direct product, (ii) the induced map g is an element of G --> [alpha(g)] is an element of Out(L) = Aut(L) / Int(L) is an injective homomorphism, and (iii) the restrictions alpha\(N),alpha\(H) are genuine actions of the subgroups on the factor L. The pair M = L x(alpha) H superset of or equal to N = L-alpha\N (of the crossed product L x(alpha) H and the fixed-point algebra L-alpha\N) gives us an irreducible inclusion of factors with Jones index #G. The inclusion M superset of or equal to N is of depth 2 and hence known to correspond to a Kac algebra of dimension #G. A Kac algebra arising in this way is investigated in detail, and in fact the relevant multiplicative unitary (satisfying the pentagon equation) is described. We introduce and analyze a certain cohomology group (denoted by H-2 ((N, H), T)) providing complete information on the Kac algebra structure, and we construct an abundance of non-trivial examples by making use of various cocycles. The operator algebraic meaning of this cohomology group is clarified, and some related topics are also discussed. Sector technique enables us to establish structure results for Kac algebras with certain prescribed underlying algebra structure. They guarantee that "most" Kac algebras of low dimension (say less than 60) actually arise from inclusions of the form L x(alpha) H superset of or equal to L-alpha\N, and consequently their classification can be carried out by determining H-2((N, H), T). Among other things we indeed classify Kac algebras of dimension 16 and 24, which (together with previously known results) gives rise to the complete classification of Kac algebras of dimension up to 31. Partly to simplify classification procedure and hopefully for its own sake, we also study "group extensions" of general (finite-dimensional) Kac algebras with some discussions on related topics.