# Kac Algebras Arising from Composition of Subfactors: General Theory and Classification, Volume 750 (Google eBook)

American Mathematical Soc., Jun 3, 2002 - Mathematics - 198 pages
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.

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### Contents

 Introduction 1 Actions of matched pairs 5 Cocycles attached to the pentagon equation 21 Multiplicative unitary 31 Kac algebra structure 37 Grouplike elements 43 Examples of finitedimensional Kac algebras 49 Inclusions with the CoxeterDynkin graph D and the KacPaljutkin algebra 63
 Structure theorems 71 Classification of certain Kac algebras 83 Classification of Kac algebras of dimension 16 107 Group extensions of general Kac algebras 123 2cocycles of Kac algebras 141 Classification of Kac algebras of dimension 24 159 Copyright