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

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 |

### Common terms and phrases

2-cocycle abelian group action of G algebra structure algebras of dimension assume automorphism Chapter closed under conjugation co-commutative coboundary cocycle cohomology group contradiction crossed product decomposed define denote depth 2 inclusion dihedral group dimension 16 dual group dual Kac algebra equation equivalent ergodic action f f i M2(C factors corresponding finite group finite-dimensional Kac algebra fixed-point fixed-point algebra fixed-point subalgebra fixed-point subgroup follows Frobenius reciprocity give rise group algebra group G group of order group ring hence Hom(N Hom(N,T Hopf algebras implies inclusion map inclusion of factors intermediate subfactor intrinsic group G(A invariant irreducible decomposition Kac-Paljutkin algebra Let G Math multiplicative unitary N M H non-trivial Kac algebra normal Note outer action principal graph projective representation PROOF rank-one satisfying self-dual semi-direct product semi-direct product group simple component thanks to Lemma trivial underlying algebra structure Xa Z2 Z2 action Z2 x Z2 Zn x Zn