CHAPTER 2
Actions of matched pairs
Let G be a finite group with subgroups Gi,G2, and we assume that G is a
product group G G\ G2. In such a case, we will call ( G i , ^ ) a matched pair.
Throughout, let 1Z be a factor.
DEFINITION 2.1.
(i) A pair of actions a : G\ Aut(lZ) o.nd (3 : G2 Aut(lZ) is called an action
of the matched pair (Gi,G2) z/ a,/? #zie G-kernel (i.e., a homomorphism from G
to Out(lZ) = Aut(lZ)/Int(lZ).
(ii) An action (a, /?) 0/ £foe matched pair (Gi, G2) ^s ow£er z/a, /? are on^er actions
of Gi,G2 respectively and the associated G-kernel is injective. Recall that this G-
kernel or rather a map G ~ G\ G2 Aut{lZ) is called a near action in [32].
(Hi) Two actions (a,/?), (a',/3') are cocycle conjugate if there are an automorphism
6 G Aut(lZ), an a-cocycle
{ ^ J ^ G G I ?
and a f3-cocycle {^g2}g2eG2 (i-e., v*gig' =
u9iagAugO andv929,2 = v92pg2(vg'2))
satisfying
Oa'g^1 = Ad(u9l)a9ll
O^O-1
= Ad(v92)f392.
In this chapter we will study various cocycles arising from an outer action of a
matched pair (Gi, G2) (in the semi-direct product case G = N x\ H) and introduce
a certain cohomology group
H2((N,H),T).
(See [21, 22] for related results.) We
will show that this cohomology group is a complete invariant for cocycle conjugacy
classes of outer actions of (]V, H) when 1Z is the hyperfinite II\ factor (Theorem
2.5). In the next chapter the cohomology group if2 ((TV, H), T) will be also identified
with the one that naturally arises from the multiplicative unitary described in the
appendix to [1].
Remarks.
1. When (a,/3) is an outer action of the matched pair (Gi,G2), by the standard
argument of bimodules or equivalently sectors we see thatftxi
a
G i Dn(G2® is an
irreducible inclusion of factors of depth 2.
2. When (a,/?) and (af, ft) are cocycle conjugate, the two inclusions 1Z xia G\ ~D
Tl(0,G2)
anc
i ft
Xa
, G1 D K^'^ are conjugate.
In fact, let w be the canonical implementation (see [67]) of #, we consider the
unitary U on L2(1Z) g £2(d) defined by (U()(gi) = u^.w^g^. Then, Ad(U)
sends 1Z x
a
/ G\ onto 1Z xa Gi, and 1Z onto itself in such a way that the restriction
to 1Z (C 1Z xa/ Gi) is 0. Therefore, Ad(U) sends
1Z^'^G^
onto the subfactor
{x e7Z : Ad(vg2)/3g2(x) = x}. Since
{V92}92^G2
is a /3-cocycle, one finds a unitary
V in 1Z satisfying v92 V*f392(V) (see [4, 6]) and the above condition means
5
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