CHAPTER 1

Introduction

The Jones index theory ([36]) deals with inclusions Af C M of factors with

finite index, and in related subfactor analysis the "position" of a subfactor J\f in

the ambient factor M. is of fundamental importance. Detailed account as well as

the state of the art of the theory can be found in the recent textbook [16] by Evans

and Kawahigashi. It is known that bimodules and sectors (see [5, 25, 46, 47] and

also the references of [16] for more detailed information) play important roles in this

subject matter. On the other hand, Hopf algebras or more precisely Kac algebras

appear in many places of the theory of operator algebras as the right framework to

handle duality phenomenon (see [13, 57] for example). Roughly speaking, a Kac

algebra means a *-Hopf algebra A (with the coproduct V : A — A ® A) equipped

with an antipode n (which is an involutive anti-automorphism of A) compatible

with the *-operation. A finite group G give us the most typical (but somewhat

simple-minded) examples of (finite-dimensional) Kac algebras: the algebra £°°(G)

of C-valued functions and the group ring C(G). Here, T,ft (for the former) and

T, k (for the latter) are given by

{TU)){g,h) = f{gh), W))(g) =

f(g'1),

t(Xg) = \g® Xg, k(Xg) = A^-l .

(Here, A (8) A is identified with £°°(G x G), the functions of two varibles, for A =

£°°(G).) The former is referred to as a commutative Kac algebra while the latter is

known as a cocommutative Kac algebra, and these two types altogether are called

trivial Kac algebras. At an early stage of the index theory it was observed by

Ocneanu that an irreducible inclusion J\f C M is of depth 2 if and only if M is

the crossed product of M relative to a coaction of a Kac algebra (of dimension

equal to the Jones index [M : A/]). Proofs were supplied by several authors ([9,

27, 29, 48, 62, 70]), and bimodules and sectors of course played important roles.

This characterization in particular means that the relevant Kac algebra structure

(together with a coaction) can be recovered from the position of J\f inside of M,

and makes it possible to use subfactor analysis for study of finite-dimensional Kac

algebras.

Assume that an outer action of a finite group G on a factor C is given. When G

is a semi-direct product G = N x\H (or more generally a product group G = N-H),

by considering the crossed product M — CKH and the fixed-point algebra Af =

CN

we get the two-step inclusion

Received by the editor September 1, 2000

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