eBook ISBN:  9781470403430 
Product Code:  MEMO/158/750.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 
eBook ISBN:  9781470403430 
Product Code:  MEMO/158/750.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 158; 2002; 198 ppMSC: Primary 46; Secondary 16
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.
ReadershipGraduate students and research mathematicians interested in functional analysis.

Table of Contents

Chapters

1. Introduction

2. Actions of matched pairs

3. Cocycles attached to the pentagon equation

4. Multiplicative unitary

5. Kac algebra structure

6. Grouplike elements

7. Examples of finitedimensional Kac algebras

8. Inclusions with the CoxeterDynkin graph $D^{(1)}_6$ and the KacPaljutkin algebra

9. Structure theorems

10. Classification of certain Kac algebras

11. Classification of Kac algebras of dimension 16

12. Group extensions of general Kac algebras

13. 2cocycles of Kac algebras

14. Classification of Kac algebras of dimension 24


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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 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.
Graduate students and research mathematicians interested in functional analysis.

Chapters

1. Introduction

2. Actions of matched pairs

3. Cocycles attached to the pentagon equation

4. Multiplicative unitary

5. Kac algebra structure

6. Grouplike elements

7. Examples of finitedimensional Kac algebras

8. Inclusions with the CoxeterDynkin graph $D^{(1)}_6$ and the KacPaljutkin algebra

9. Structure theorems

10. Classification of certain Kac algebras

11. Classification of Kac algebras of dimension 16

12. Group extensions of general Kac algebras

13. 2cocycles of Kac algebras

14. Classification of Kac algebras of dimension 24