# Kac Algebras Arising from Composition of Subfactors: General Theory and Classification

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*Masaki Izumi; Hideki Kosaki*

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.

#### Table of Contents

# Table of Contents

## Kac Algebras Arising from Composition of Subfactors: General Theory and Classification

- Contents vii8 free
- Chapter 1. Introduction 112 free
- Chapter 2. Actions of matched pairs 516 free
- Chapter 3. Cocycles attached to the pentagon equation 2132
- Chapter 4. Multiplicative unitary 3142
- Chapter 5. Kac algebra structure 3748
- Chapter 6. Group-like elements 4354
- Chapter 7. Examples of finite-dimensional Kac algebras 4960
- Chapter 8. Inclusions with the Coxeter-Dynkin graph D[sup((1))][sub(6)] and the Kac-Paljutkin algebra 6374
- Chapter 9. Structure theorems 7182
- Chapter 10. Classification of certain Kac algebras 8394
- Chapter 11. Classification of Kac algebras of dimension 16 107118
- Chapter 12. Group extensions of general Kac algebras 123134
- Chapter 13. 2-cocycles of Kac algebras 141152
- Chapter 14. Classification of Kac algebras of dimension 24 159170
- Bibliography 193204
- Index 196207 free