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

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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