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Kac Algebras Arising from Composition of Subfactors: General Theory and Classification
 
Masaki Izumi Kyoto University, Kyoto, Japan
Hideki Kosaki Kyushu University, Fukuoka, Japan
Kac Algebras Arising from Composition of Subfactors: General Theory and Classification
eBook ISBN:  978-1-4704-0343-0
Product Code:  MEMO/158/750.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
Kac Algebras Arising from Composition of Subfactors: General Theory and Classification
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Kac Algebras Arising from Composition of Subfactors: General Theory and Classification
Masaki Izumi Kyoto University, Kyoto, Japan
Hideki Kosaki Kyushu University, Fukuoka, Japan
eBook ISBN:  978-1-4704-0343-0
Product Code:  MEMO/158/750.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1582002; 198 pp
    MSC: 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 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
     
     
    • Chapters
    • 1. Introduction
    • 2. Actions of matched pairs
    • 3. Cocycles attached to the pentagon equation
    • 4. Multiplicative unitary
    • 5. Kac algebra structure
    • 6. Group-like elements
    • 7. Examples of finite-dimensional Kac algebras
    • 8. Inclusions with the Coxeter-Dynkin graph $D^{(1)}_6$ and the Kac-Paljutkin 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. 2-cocycles of Kac algebras
    • 14. Classification of Kac algebras of dimension 24
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1582002; 198 pp
MSC: 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 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.

  • Chapters
  • 1. Introduction
  • 2. Actions of matched pairs
  • 3. Cocycles attached to the pentagon equation
  • 4. Multiplicative unitary
  • 5. Kac algebra structure
  • 6. Group-like elements
  • 7. Examples of finite-dimensional Kac algebras
  • 8. Inclusions with the Coxeter-Dynkin graph $D^{(1)}_6$ and the Kac-Paljutkin 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. 2-cocycles of Kac algebras
  • 14. Classification of Kac algebras of dimension 24
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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