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Kac Algebras Arising from Composition of Subfactors: General Theory and Classification
Masaki Izumi, Kyoto University, Japan, and Hideki Kosaki, Kyushu University, Fukuoka, Japan
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Memoirs of the American Mathematical Society
2002; 198 pp; softcover
Volume: 158
ISBN-10: 0-8218-2935-1
ISBN-13: 978-0-8218-2935-6
List Price: US$67
Individual Members: US$40.20
Institutional Members: US$53.60
Order Code: MEMO/158/750
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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.

Table of Contents

  • Introduction
  • Actions of matched pairs
  • Cocycles attached to the pentagon equation
  • Multiplicative unitary
  • Kac algebra structure
  • Group-like elements
  • Examples of finite-dimensional Kac algebras
  • Inclusions with the Coxeter-Dynkin graph \(D^{(1)}_6\) and the Kac-Paljutkin algebra
  • Structure theorems
  • Classification of certain Kac algebras
  • Classification of Kac algebras of dimension 16
  • Group extensions of general Kac algebras
  • 2-cocycles of Kac algebras
  • Classification of Kac algebras of dimension 24
  • Bibliography
  • Index
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