Memoirs of the American Mathematical Society 2002; 198 pp; softcover Volume: 158 ISBN10: 0821829351 ISBN13: 9780821829356 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/158/750
 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 semidirect 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 fixedpoint 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 nontrivial 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 (finitedimensional) 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
 Grouplike elements
 Examples of finitedimensional Kac algebras
 Inclusions with the CoxeterDynkin graph \(D^{(1)}_6\) and the KacPaljutkin algebra
 Structure theorems
 Classification of certain Kac algebras
 Classification of Kac algebras of dimension 16
 Group extensions of general Kac algebras
 2cocycles of Kac algebras
 Classification of Kac algebras of dimension 24
 Bibliography
 Index
