We deal with a map $\alpha$ from a finite group $G$ into the
automorphism group $Aut({\mathcal L})$ of a factor ${\mathcal
L}$ satisfying (i) $G=N \rtimes H$ is a semi-direct product, (ii)
the induced map $g \in G \to [\alpha_g] \in Out({\mathcal L})=Aut({\mathcal
L})/Int({\mathcal L})$ is an injective homomorphism, and (iii) the
restrictions $\alpha \! \! \mid_N, \alpha \! \! \mid_H$ are genuine
actions of the subgroups on the factor ${\mathcal L}$. The pair
${\mathcal M}={\mathcal L} \rtimes_{\alpha} H \supseteq {\mathcal
N}={\mathcal L}^{\alpha\mid_N}$ (of the crossed product ${\mathcal L}
\rtimes_{\alpha} H$ and the fixed-point algebra ${\mathcal
L}^{\alpha\mid_N}$) gives us an irreducible inclusion of factors with
Jones index $\# G$. The inclusion ${\mathcal M} \supseteq
{\mathcal N}$ 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
$H^2((N,H),{\mathbf T})$) 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 ${\mathcal L} \rtimes_{\alpha} H
\supseteq {\mathcal L}^{\alpha\mid_N}$, and consequently their
classification can be carried out by determining $H^2((N,H),{\mathbf
T})$. 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.

Readership

Graduate students and research mathematicians interested in
functional analysis.