Quantum group-twisted tensor products of C*-algebras II

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by  Ralf Meyer

QUANTUM GROUP-TWISTED TENSOR PRODUCTS OF
C∗-ALGEBRAS II
RALFMEYER,SUTANUROY,ANDSTANISŁAWLECHWORONOWICZ
Abstract. For aquasitriangular C∗-quantum group, weenrich the twisted
5 tensorproductconstructedinthefirstpartofthisseriestoamonoidalstruc-
1 ture on the category of its continuous coactions on C∗-algebras. We define
0 braided C∗-quantum groups, where the comultiplication takes values in a
2 twistedtensor product. Weshow thatcompact braidedC∗-quantum groups
yieldcompactquantum groupsbyasemidirectproductconstruction.
n
u
J

1

] 1. Introduction
A
O Let C and D be C∗-algebras with a coaction of a C∗-quantum group G =
(A,∆ ). As in [12], C∗-quantum groups are generated by manageable multiplica-
. A
h tive unitaries, and Haar weights are not assumed. If G is a group, then the
at C∗-tensor product C ⊗D inherits a diagonal coaction. This fails for quantum
m groups because the diagonal coaction is not compatible with the multiplication
[ in the tensor product. We use the noncommutative tensor products described
in [12] to construct a monoidal structure on the category of G-C∗-algebrasif G is
2
quasitriangular in a suitable sense.
v
Sucha structure is to be expected from the analogoussituation for (co)module
2
3 algebrasovera Hopfalgebra. Inthat context,anR-matrixforthe dualHopfalge-
4 bra allows to deform the multiplication on the tensor product of two H-comodule
4 algebrassoastogetanH-comodulealgebraagain. ForC∗-quantumgroups,Hopf
0 module structures are replaced by comodule structures. Hence we call G quasi-
.
1 triangular if there is a unitary R-matrix R ∈ U(Aˆ⊗Aˆ) for the dual C∗-quantum
0
group.
5
Since R is a bicharacter, the braided tensor product C⊠D :=(C,γ)⊠ (D,δ)
1 R
∗
: in [12] is defined if (C,γ) and (D,δ) are C -algebras with continuous coactions
v of G. We show that C ⊠D carries a unique continuous coaction γ ⊲⊳ δ of G for
i
X whichthecanonicalembeddingsofC andD areG-equivariant. (Wedonotdenote
r this coaction by γ⊠δ because ⊠ is a bifunctor, and the *-homomorphism γ⊠δ
a
given by this bifunctoriality is not γ ⊲⊳δ.)
2010 Mathematics Subject Classification. 46L55(81R50,46L06, 18D10).
Keywordsandphrases. C∗-quantumgroup,braiding,quantumcodouble,semidirectproduct,
bosonisation.
Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG))
throughtheResearchTrainingGroup1493andbytheAlexandervonHumboldt-Stiftung.
1
2 RALFMEYER,SUTANUROY,ANDSTANISŁAWLECHWORONOWICZ
Itiscrucialforthetheoryhereandin[12]thatRisunitary. Thisrulesoutsome
importantexamplesofquasitriangularHopfalgebras. Forinstance,R-matricesfor
quantum deformations of compact simple Lie groups are non-unitary.
If (E,ǫ) is another C∗-algebra with a continuous coaction of G, then there is a
canonical isomorphism (C ⊠D)⊠E ∼= C ⊠(D⊠E). If C or D carries a trivial
G-coaction,thenC⊠D=C⊗D,andγ ⊲⊳δ istheobviousinducedaction,γ⊗id
D
or id ⊗δ. Thus our tensor product on G-coactions is monoidal: the tensor unit
C
is C with trivial coaction. The tensor product of coactions is braided monoidal if
andonly if it is symmetric monoidal,if andonly if the R-matrix is antisymmetric.
This rarely happens, and it should not be expected because this also usually fails
ontheHopfalgebralevel. WhatshouldbebraidedisthecategoryofHilbertspace
corepresentations. This is indeed the case, and we use it to prove that the tensor
product for coactions is associative and monoidal.
An R-matrix R∈U(Aˆ⊗Aˆ) lifts uniquely to a universal R-matrix R∈U(Aˆu⊗
Aˆu) for the universal quantum group Aˆu, so it makes no difference whether we
consider R-matrices for Aˆ or Aˆu. Since Hilbert space corepresentations of A are
equivalent to Hilbert space representations of Aˆu, an R-matrix for Aˆu induces a
braiding on the monoidal category of Hilbert space corepresentations of G.
If G is the quantum groupof functions on an Abelian locally compactgroupΓ,
then its R-matrices are simply bicharacters Γˆ × Γˆ → U(1). For instance, if Γ
is Z/2, there are two such bicharacters. One gives the ordinary commutative
tensor product with the diagonal coaction, the other gives the skew-commutative
tensor product with diagonal Z/2-coaction.
A well-knownclass of quasitriangularHopf algebrasare Drinfeld doubles; their
modulealgebrasarethesameasYetter–Drinfeldalgebras. ThedualoftheDrinfeld
double D(G) of G is the Drinfeld codouble D(G) , which we just call quantum
codouble. (What we call quantum codouble is called quantum double in [13].)
ForourclassofC∗-quantumgroups,quantumcobdoublesanddoublesandcorre-
sponding multiplicative unitaries are described in [17]. It is already shown in [17]
that quantum codoubles are quasitriangular and that D(G) -C∗-algebras are the
sameasG-Yetter–DrinfeldC∗-algebras. Inthisarticle,weidentifythetwistedten-
sor product for the canonical R-matrix of D(G) with the twbisted tensor product
in the category of G-Yetter–Drinfeld C∗-algebras constructed in [13].
A braided C∗-bialgebra is a G-C∗-algebra (B,bβ) with a comultiplication
∆ : B →B⊠B
B
that is coassociative. We call (B,β,∆ ) a braided compact quantum group if B is
B
unital and ∆ satisfies an appropriate Podleś condition.
B
WeareparticularlyinterestedinbraidedquantumgroupsoveracodoubleD(G)
because they appear in a quantum group version of semidirect products. In the
groupcase,theconstructionofasemidirectproductG⋉H requiresaconjugationb
action of G on H such that the multiplication map H ×H →H is G-equivariant.
For quantum group semidirect products, the equivariance of the multiplication
map on the underlying C∗-algebra B of H only makes sense if we deform the ten-
sor product because there is no canonical G-coaction on B ⊗B. A theorem of
Radford[15,Theorem3]forthe analogoussituationinthe worldofHopf algebras
QUANTUM GROUP-TWISTED TENSOR PRODUCTS OF C∗-ALGEBRAS II 3
suggests that H should be a braided quantum group over the codouble D(G)
of G. In this case, we describe an induced C∗-bialgebra structure on A⊠B. This
∗ b
isaC -algebraicanalogueofwhatMajidcalls“bosonisation”in[10]. Wepreferto
callthe constructionofA⊠B a “semidirect product.” If A is a compactquantum
group and B is a braided compact quantum group, then their semidirect product
A⊠B is a compact quantum group.
As a first example, we construct a C∗-algebraic analogue of the partial duals
studied in [3]. We have constructed braided quantum SU(2) groups with complex
deformation parameter q together with Paweł Kasprzak in [6]; their semidirect
products are the deformation quantisations of the unitary group U(2) defined
in [22].
The construction of the C∗-bialgebra A⊠B works in greatgenerality. For this
to be a C∗-quantumgroup,we wouldneed a multiplicative unitary for it. Then it
is best to work on the level of multiplicative unitaries throughout. That is done
in [16]. On that level, one can also go back and decompose a semidirect product
into the two factors. Here we limit our attention to the compact case, where
bisimplifiability is enough to get a quantum group.
Webrieflysummarisethefollowingsections. InSection2,wedefine R-matrices
andshow thatthey lift to the universalquantum group. InSection 3, we describe
the braidedmonoidalstructure onthe categoryofHilbertspacecorepresentations
for a quasitriangular C∗-quantum group. As an example, we consider the case of
Abeliangroups. InSection4we constructthe “diagonal”actionofaquasitriangu-
lar quantum group on tensor products twisted by the R-matrix and show that it
givesamonoidalstructureonG-C∗-algebras. Section5studiesthecaseofquantum
∗
codoubles, where coactions on C -algebras are equivalent to Yetter–Drinfeld alge-
brastructures. Section6containsthe semidirectproductconstructionforbraided
∗ ∗
C -bialgebras. The appendix recalls basic results about C -quantum groups and
some results of our previous articles for the convenience of the reader. There are
alsosomenewobservationsaboutHeisenbergpairsinAppendix A.7,whichwould
fit better into [12] but were left out there.
2. R-matrices
LetG=(A,∆ )be a C∗-quantumgroupandletW ∈U(Aˆ⊗A)be its reduced
A
bicharacter; see Appendix A.1 and Definition A.17.
Definition 2.1. A bicharacter R∈U(A⊗A) is called an R-matrix if
(2.2) R(σ◦∆ (a))R∗ =∆ (a) for all a∈A.
A A
Lemma 2.3. The dual Rˆ :=σ(R∗)∈U(A⊗A) of a bicharacter R∈U(A⊗A) is
an R-matrix if and only if R is an R-matrix. (cid:3)
Remark 2.4. The standard convention for Hopf algebras assumes R(∆ (a))R∗ =
A
σ◦∆ (a),whichisoppositeto(2.2). Ourconventionin2.2becomesthe standard
A
one (see [9, Definition 2.1.1]) if we replace ∆ by ∆cop :=σ◦∆ or R by R∗.
A A A
Inordertosimplifyproofslater,weliftanR-matrixR∈U(A⊗A)toU(Au⊗Au):
4 RALFMEYER,SUTANUROY,ANDSTANISŁAWLECHWORONOWICZ
Proposition 2.5. There is a unique Ru ∈U(Au⊗Au) with
(2.6) (Λ⊗Λ)Ru =R in U(A⊗A),
(2.7) (∆Au ⊗idAu)Ru =Ru23Ru13 in U(Au⊗Au⊗Au),
(2.8) (idAu ⊗∆Au)Ru =Ru12Ru13 in U(Au⊗Au⊗Au).
This unitary also satisfies
(2.9) Ru(σ◦∆u(a))(Ru)∗ =∆u(a) for all a∈Au.
A A
Proof. [11, Proposition4.7]givesaunique Ru ∈U(Au⊗Au)satisfying (2.6)–(2.8).
The nontrivial part is to show that Ru satisfies (2.9). Let V ∈ U(Aˆ⊗Au) be the
universal bicharacter as in Appendix A.4. Theorem 25 and Proposition31 in [19]
show that
(2.10) Au ={(ω⊗idAu)V |ω ∈Aˆ′}CLS and (idAˆ⊗∆Au)V =V12V13.
Therefore, (2.9) is equivalent to:
(2.11) Ru V V (Ru )∗ =V V in U(Aˆ⊗Au⊗Au).
23 13 12 23 12 13
The unitary R := (Λ⊗idAu)Ru ∈ M(A⊗Au) is also a bicharacter. Let X :=
W∗ R V W ∈ U(Aˆ⊗A⊗Au). The following computation shows that X is a
12 23 13 12
corepresentationof (Au,∆Au) on Aˆ⊗A:
∗ ∗
(idAˆ⊗idA⊗∆Au)(W12R23V13W12)=W12R23R24V13V14W12
∗ ∗
=(W R V W )(W R V W )=X X .
12 23 13 12 12 24 14 12 123 124
The first step uses (2.8) and (2.10), the second step uses that R and V com-
24 13
mute, and the last step is trivial. A similar routine computation shows that
Y :=V13R23 ∈U(Aˆ⊗A⊗Au) satisfies (idAˆ⊗idA⊗∆Au)Y =Y123Y124.
The argument that shows that (2.9) is equivalent to (2.11) also shows that the
R-matrix condition (2.2) is equivalent to
(2.12) R W W =W W R in U(Aˆ⊗A⊗A).
23 13 12 12 13 23
Thus (id ⊗id ⊗Λ)X =(id ⊗id ⊗Λ)Y. Nowwe use Lemma [11, Lemma 4.6],
Aˆ A Aˆ A
which is a variation on [7, Result 6.1]. It gives X =Y or, equivalently,
(2.13) V∗ W∗ R¯ V =R¯ W∗ in U(Aˆ⊗A⊗Au).
13 12 23 13 23 12
Similarly, X := V∗ (Ru )∗V V and Y := V (Ru )∗ in U(Aˆ⊗Au ⊗Au) satisfy
13 23 12 13 12 23
(idAˆ⊗Λ⊗eidAu)X =(idAˆ⊗Λ⊗idAu)Ye by (2.13), and
(idAˆ⊗∆Au ⊗iedAu)(X)=X124X134,e (idAˆ⊗∆Au ⊗idAu)(Y)=Y124Y134
because of (2.7) and (2.e10). Aenotheer application of [11, Lemma 4e.6]givees Xe =Y,
which is equivalent to (2.11). (cid:3)
e e
[19, Proposition 31.2] shows that (Au,∆Au) has a bounded counit: there is a
unique morphism e: Au →C with
(2.14) (e⊗idAu)∆Au =(idAu ⊗e)∆Au =idAu.
QUANTUM GROUP-TWISTED TENSOR PRODUCTS OF C∗-ALGEBRAS II 5
Lemma 2.15. The unitary Ru ∈U(Au⊗Au) in Proposition 2.5 satisfies
(2.16) (e⊗idAu)Ru =(idAu ⊗e)Ru =1Au in U(Au),
(2.17) Ru Ru Ru =Ru Ru Ru in U(Au⊗Au⊗Au).
12 13 23 23 13 12
Proof. Apply idAu⊗e⊗idAu on both sides of (2.7) and (2.8) and then use (2.14).
This gives
Ru =(1Au ⊗(e⊗idAu)Ru)Ru =(((idAu ⊗e)Ru)⊗1Au)Ru.
Multiplying with (Ru)∗ on the right gives (e⊗idAu)Ru =(idAu ⊗e)Ru =1Au.
The following computation yields (2.17):
Ru12Ru13Ru23 =((idAu ⊗∆u)Ru)Ru23 =Ru23((idAu ⊗σ◦∆u)Ru)=Ru23Ru13Ru12;
here the first and third step use (2.8) and the second step uses (2.9). (cid:3)
3. Corepresentation categories of quasitriangular quantum groups
Definition 3.1. A quasitriangular C∗-quantum group is a C∗-quantum group
G=(A,∆ ) with an R-matrix R∈U(Aˆ⊗Aˆ).
A
LetΣ(H1,H2): H ⊗H →H ⊗H denotetheflipoperator. Asalreadypointed
1 2 2 1
outin[19],Σ(H1,H2) isG-equivariantforallcorepresentationsofGifandonlyifG
12
is commutative. Hence Σ(·,·) does not give a braiding on Corep(G) in general.
Let UHi ∈ U(K(H )⊗A) be corepresentations of G on H for i = 1,2. These
i i
correspondtorepresentationsofthe universalquantumgroupAˆu bythe universal
property of Aˆu. More precisely, there are unique ϕˆ ∈ Mor(Aˆu,K(H )) such that
i i
(ϕˆ ⊗ id )V˜ = UHi for i = 1,2, see Appendix A.4; here V˜A is the universal
i A
bicharacter in U(Aˆu⊗A).
Define H1 H2: H ⊗H →H ⊗H by
1 2 2 1
(3.2) X(H2,H1) :=(ϕˆ ⊗ϕˆ )(Ru)∗ in U(H ⊗H ),
2 1 2 1
(3.3) H1 H2 :=X(H2,H1)◦ΣH1,H2 in U(H ⊗H ,H ⊗H ).
1 2 2 1
Here Ru ∈U(Aˆu⊗Aˆu) is as in Proposition 2.5.
Proposition 3.4. The unitaries H1 H2: H ⊗H →H ⊗H are G-equivariant,
1 2 2 1
that is,
(3.5) H1 H2 (UH1 UH2)=(UH2 UH1)H1 H2 in U(K(H ⊗H )⊗A)
12 12 1 2
for all UH1,UH2 ∈Corep(G). The tensor product is defined in (A.12).
6 RALFMEYER,SUTANUROY,ANDSTANISŁAWLECHWORONOWICZ
TheunitariesH1 H2 defineabraidingonCorep(G),thatis,thefollowinghexagons
commute for all UHi ∈Corep(G), i=1,2,3:
H1 H2⊗H3
H ⊗(H ⊗H ) (H ⊗H )⊗H
1 2 3 2 3 1
(3.6) (H ⊗H )⊗H H ⊗(H ⊗H )
1 2 3 2 3 1
H1 H2⊗idH3 idH2 ⊗H1 H3
(H ⊗H )⊗H H ⊗(H ⊗H )
2 1 3 2 1 3
H1⊗H2 H3
(H ⊗H )⊗H H ⊗(H ⊗H )
1 2 3 3 1 2
(3.7) H ⊗(H ⊗H ) (H ⊗H )⊗H
1 2 3 3 1 2
idH1 ⊗H2 H3 H1 H3⊗idH2
H ⊗(H ⊗H ) (H ⊗H )⊗H
1 3 2 1 3 2
Here the unlabelled arrows are the standard associators of Hilbert spaces.
Proof. We have (∆ˆAu ⊗idA)V˜A = V˜2A3V˜1A3 in U(Aˆu ⊗Aˆu ⊗A) because V˜A is a
character in the first leg. Therefore, the corepresentationUH1 UH2 corresponds
to (φˆ ⊗φˆ )◦σ◦∆ˆu : Aˆu → B(H ⊗H ) through the universal property (A.16)
1 2 A 1 2
of V˜:
(3.8) ((φˆ ⊗φˆ )◦σ◦∆ˆu ⊗id )V˜ =UH1 UH2.
1 2 A A
The following computation yields (3.5):
H1 H2 (UH1 UH2)=(φˆ ⊗φˆ ⊗id ) (Ru )∗(∆ˆu ⊗id )V˜ Σ(H1,H2)
12 2 1 A 12 A A 12
(cid:0) (cid:1)
= ((φˆ ⊗φˆ )◦σ◦∆ˆu)⊗id )V˜ X(H2,H1)Σ(H1,H2)
2 1 A A 12 12
(cid:0) (cid:1)
=(UH2 UH1)H1 H2 .
12
The first equality uses (3.8) and (3.3), the second equality follows from (2.9) and
(3.2), and the last equality uses (3.8) and (3.3).
Equations (3.3) and (3.8) imply
(3.9) H1 H2⊗H3 :=X(H2⊗H3,H1)Σ(H1,H2⊗H3)
= (φˆ ⊗φˆ ⊗φˆ )(σ◦∆ˆu ⊗id )(Ru)∗ ◦Σ(H1,H2⊗H3).
2 3 1 A Aˆu
(cid:16) (cid:17)
QUANTUM GROUP-TWISTED TENSOR PRODUCTS OF C∗-ALGEBRAS II 7
Now we check the first braiding diagram (3.6):
(φˆ ⊗φˆ ⊗φˆ )(σ◦∆ˆu ⊗id )(Ru)∗ ◦Σ(H1,H2⊗H3)
2 3 1 A Aˆu
(cid:16) (cid:17)
= (φˆ ⊗φˆ ⊗φˆ ) (Ru)∗ (Ru)∗ Σ(H1,H3)Σ(H1,H2)
2 3 1 23 13 23 12
(cid:16) (cid:0) (cid:1)(cid:17)
=X(H3,H1)X(H2,H1)Σ(H1,H3)Σ(H1,H2)
23 13 23 12
=X(H3,H1)Σ(H1,H3)X(H2,H1)Σ(H1,H2) =H3 H1 H2 H1 ;
23 23 12 12 23 12
here the first equality uses (2.7), the second equality uses (3.2), the third equality
uses properties of the flip operator Σ, and the fourth equality follows from (3.3).
A similar computation for H1⊗H2 H3 yields the second braiding diagram (3.7).
(cid:3)
Corollary 3.10. If C carries the trivial corepresentation of G, then
C H: C⊗H→H⊗C and H C: H⊗C→C⊗H
are the canonical isomorphisms. For any three corepresentations of G,
(3.11) H1 H2 H1 H3 H2 H3 =H2 H3 H1 H3 H1 H2 .
23 12 23 12 23 12
Proof. These are general properties of braided monoidal categories,see [5, Propo-
sition 2.1]. They also follow from (2.16), (2.17), and (3.2). (cid:3)
Remark 3.12. The dual Rˆ := σ(R∗) of an R-matrix R ∈ U(Aˆ⊗Aˆ) is again an
R-matrixbyLemma2.3. Aroutinecomputationshowsthattheresultingbraiding
on Corep(G) is the dual braiding, given by the braiding unitaries
H1 H2 = H2 H1 ∗: H ⊗H →H ⊗H .
1 2 2 1
(cid:0) (cid:1)
3.1. Symmetric braidings.
Definition3.13. AnR-matrixR∈U(A⊗A)iscalledantisymmetricifR∗ =σ(R)
for the flip σ: A⊗A→A⊗A, a ⊗a 7→a ⊗a .
1 2 2 1
Lemma 3.14. If R is antisymmetric, then (Ru)∗ = σ(Ru) for the universal lift
Ru ∈U(Au⊗Au) constructed in Proposition 2.5.
Proof. Both σ(Ru)∗ and Ru are bicharacters that lift R. They must be equal
because bicharacters lift uniquely by [11, Proposition 4.7]. (cid:3)
Proposition 3.15. The braiding on Corep(G) constructed from R∈U(Aˆ⊗Aˆ) is
symmetric if and only if R is antisymmetric.
Proof. LetH andH beHilbertspaceswithcorepresentationsofG. Letφˆ : Aˆu →
1 2 i
B(H ) be the corresponding *-representations. Then
i
H1 H2 H2 H1
H ⊗H −−−−−→H ⊗H −−−−−→H ⊗H
1 2 2 1 1 2
is equal to
(φˆ ⊗φˆ )(Ru)∗◦Σ(H2,H1)◦(φˆ ⊗φˆ )(Ru)∗◦Σ(H1,H2) =(φˆ ⊗φˆ )(σ(Ru)Ru)∗.
1 2 2 1 1 2
This is the identity operator for all representations φˆ if and only if σ(Ru)Ru =
i
1. (cid:3)
8 RALFMEYER,SUTANUROY,ANDSTANISŁAWLECHWORONOWICZ
3.2. The Abeliancase. LetB bealocallycompactgroup. WhatisanR-matrix
forthe commutativequantumgroup(C (G),∆)? Since C (G)⊗C (G) iscommu-
0 0 0
tativeaswell,(2.2)simplifiestotheconditionσ◦∆ =∆,whichisequivalenttoG
being commutative. Hence there is no R-matrix unless G is Abelian, which we as-
sume from now on. Then (2.2) holds for any unitary R ∈ U(C (G)⊗ C (G)),
0 0
so an R-matrix for G is simply a bicharacter. Equivalently, R is a function
ρ: G×G→U(1) satisfying ρ(xy,z)= ρ(x,z)ρ(y,z) and ρ(x,yz)=ρ(x,y)ρ(x,z).
Being antisymmetric means ρ(x,y)ρ(y,x)=1 for all x,y ∈G.
Any bicharacter ρ as above is of the form ρ(x,y)=hρˆ(x),yi for a group homo-
morphism ρˆ: G → Gˆ to the Pontrjagin dual Gˆ, with ρ(x,␣) = ρˆ. This is a spe-
cial case of the interpretation of bicharacters as quantum group homomorphisms
in [11].
The category of Hilbert space representations of G is equivalent to the cate-
gory of corepresentations of (C (G),∆) and to the category of representations of
0
C∗(G) ∼= C (Gˆ). The tensor category of G-representations is already symmetric
0
for the obvious braiding Σ, which corresponds to the R-matrix 1. What are the
braiding operators for a nontrivial R-matrix?
Let ⊕H dµ(x) denote the Hilbert space of L2-sections of a measurable field
Gˆ x
of HilbRert spaces (H ) over Gˆ with respect to a measure µ, equipped with
x x∈Gˆ
the action of C (Gˆ) by pointwise multiplication. Any representation of C (Gˆ) is
0 0
of this form, where µ is unique up to measure equivalence and the field (H ) is
x
⊕
unique up to isomorphism µ-almost everywhere. Let H = H dµ (x) and
1 Gˆ 1x 1
H = ⊕H dµ (x) be two Hilbert space representations of GR. Then
2 Gˆ 2x 2
R
⊕
H ⊗H = H ⊗H dµ (x)dµ (y)
1 2 Z 1x 2y 1 2
Gˆ×Gˆ
withC (Gˆ)⊗C (Gˆ)∼=C (Gˆ×Gˆ)actingbypointwisemultiplication. Thebraiding
0 0 0
H1 H2 maps an L2-section (ξ ) of the field (H ⊗ H ) to the section
x,y x,y 1x 2y x,y
(y,x)7→ρ(y,x)−1ξ of (H ⊗H ) .
x,y 2y 1x y,x
Example 3.16. Consider G = Z/2 = {±1} and let ρ(x,y) = xy ∈ Z/2 ⊆ U(1);
this bicharacter corresponds to the isomorphism G ∼= Gˆ. It is both symmetric
and antisymmetric. The spectral analysis above writes a Z/2-Hilbert space as a
Z/2-graded Hilbert space, splitting it into even and odd elements with respect to
the action of the generator of Z/2. The braiding unitary on ξ⊗η is Σ if ξ or η is
even, and −Σ if both ξ and η are odd. This is the usual Koszul sign rule.
4. Coaction categories of quasitriangular quantum groups
Let G=(A,∆ ,R) be a quasitriangularquantum group. Let (C,γ) and (D,δ)
A
be G-C∗-algebras. The twisted tensor product C ⊠ D = C ⊠D is constructed
R
in[12]. It isa crossedproductofC andD, thatis,there arecanonicalmorphisms
ι : C →C⊠D and ι : D →C⊠D with
C D
ι (C)·ι (D)=ι (D)·ι (C)=C⊠D;
C D D C
QUANTUM GROUP-TWISTED TENSOR PRODUCTS OF C∗-ALGEBRAS II 9
here a morphism is a nondegenerate *-homomorphism to the multiplier algebra
andX·Y fortwosubspacesX andY ofaC∗-algebrameanstheclosed linear span
of x·y for x∈X, y ∈Y as in [12].
TheoremA.29recallsoneofthe twoequivalentdefinitionsofthe twistedtensor
product in [12]. Let (ϕ,UH) and (ψ,UK) be faithful covariant representations
of (C,γ) and (D,δ) on Hilbert spaces H and K, respectively. Then C ⊠ D is
R
canonicallyisomorphictoφ1(C)·ψ˜2(D)⊆B(H⊗K), whereφ1(c)=φ(c)⊗1K and
ψ˜2(d)=X(1H⊗ψ(d))X∗ for the unitary X that is characterised by (A.27). The
same unitary appears in our construction of the braiding, so
(4.1) ψ˜2(d)=H K(ψ(d)⊗1H)(H K)∗.
WearegoingtoequipC⊠DwithanaturalG-coactionandshowthatthistensor
productgivesamonoidalstructureonthecategoryofG-C∗-algebrasC∗alg(G)(see
Definition A.13).
Proposition 4.2. There is a unique G-coaction γ ⊲⊳ δ on C ⊠ D such that
R R
the canonical representation on H⊗K and the corepresentation UH UK form a
covariant representation of (C⊠ D,γ ⊲⊳ δ). This coaction is also theuniqueone
R R
forwhichthemorphismsι : C →C⊠ D andι : D →C⊠ D areG-equivariant.
C R D R
Proof. We identify C ⊠D with its image in B(H⊗K). The covariance of this
representation of C⊠D with UH UK means that
(γ ⊲⊳ δ)(x)=(UH UK)(x⊗1 )(UH UK)∗
R A
for all x∈C⊠D. Hence there is at most one such coaction γ ⊲⊳ δ.
R
H K
Therepresentationc7→ιC(c)=φ(c)⊗1K is covariantwith respectto U U
because it is covariantwith respectto UH⊗1 by constructionandι (c) acts only
C
on the first leg. Hence (γ ⊲⊳ δ)(ι (c)) = (ι ⊗id )γ(c) for all c ∈ C. Similarly,
R C C A
K H
the representation d 7→ψ(d)⊗1 on K⊗H is covariant with respect to U U .
Since the unitary H K is G-equivariant by Proposition 3.4, the representation ψ˜
2
on H⊗K is covariantwith respect to UH UK as well (unlike the representation
d7→1⊗ψ(d)). Hence (γ ⊲⊳ δ)(ι (d))=(ι ⊗id )δ(d) foralld∈D. As a result,
R D D A
γ ⊲⊳ δ maps C⊠D =ι (C)·ι (D) nondegenerately into the multiplier algebra
R C D
of (C⊠D)⊗A, and the morphisms ι and ι are G-equivariant.
C D
The morphism γ ⊲⊳ δ: C⊠D →(C⊠D)⊗A is faithful by construction. The
R
Podleś condition for C⊠D follows from those for C and D:
(γ ⊲⊳ δ)(C ⊠D)·(1⊗A)=(ι ⊗id )(γ(C))·(ι ⊗id )(δ(D))·(1⊗A)
R C A D A
=(ι ⊗id )(γ(C))·(ι (D)⊗A)=(ι ⊗id )(γ(C))·(1⊗A)·(ι (D)⊗A)
C A D C A D
=(ι (C)⊗A)·(ι (D)⊗A)=C⊠D⊗A.
C D
Thus γ ⊲⊳ δ is a continuous G-coaction on C ⊠ D for which ι and ι are
R C D
equivariant. Conversely,if ι and ι are G-equivariant,then
C D
(γ ⊲⊳ δ)(ι (c)·ι (d))=(ι ⊗id )(γ(c))·(ι ⊗id )(δ(d))
R C D C A D A
for c∈C, d∈D; this determines γ ⊲⊳ δ because ι (C)·ι (D)=C⊠D. (cid:3)
R C D
10 RALFMEYER,SUTANUROY,ANDSTANISŁAWLECHWORONOWICZ
Proposition4.3. Thecoactionγ ⊲⊳ onC⊠ Disnaturalwithrespecttoequivari-
R R
ant morphisms, that is, it gives a bifunctor ⊠ : C∗alg(G)×C∗alg(G)→C∗alg(G).
R
ItistheonlynaturalcoactionforwhichCwiththeobviousisomorphismsC⊠C∼=C
and C⊠D ∼=D is a tensor unit.
Proof. Two G-equivariant morphisms f: C → C and g: D → D induce a
1 2 1 2
morphism f ⊠ g: C ⊠ D → C ⊠ D , which is determined uniquely by the
1 1 2 2
conditions (f⊠g)◦ι =ι ◦f and(f⊠g)◦ι =ι ◦g (see [12, Lemma 5.5]).
C1 C2 D1 D2
The coactions γ ⊲⊳ δ and γ ⊲⊳ δ satisfy (γ ⊲⊳ δ )◦ι = (ι ⊗id )◦γ
1 R 1 2 R 2 k R k Ck Ck A k
and (γ ⊲⊳ δ )◦ι =(ι ⊗id )◦δ for k =1,2. Thus
k R k Dk Dk A k
(γ ⊲⊳ δ )◦ι ◦f =(f ⊠g⊗id )◦(γ ⊲⊳ δ )◦ι
2 R 2 C2 A 1 R 1 C1
andsimilarlyonD . Sof⊠gisequivariantand⊠ isabifunctorasasserted. The
1 R
obvious isomorphisms C⊠C∼=C and C⊠D ∼=D are G-equivariant and natural
and satisfy the triangle axiom for a tensor unit in a monoidalcategory;so C with
these isomorphisms is a unit for the tensor product ⊠ on C∗alg(G).
R
Conversely, assume that γ ⊡δ is a natural G-coaction on C ⊠D for which C
with the obvious isomorphisms C⊠C∼=C and C⊠D ∼=D is a tensor unit. That
is, these two isomorphisms are G-equivariant. The unique morphisms 1 : C→C
C
and 1 : C → D given by the unit multiplier are equivariant with respect to the
D
trivial G-coaction on C. We have ι = id ⊠1 and ι = 1 ⊠id . Hence ι
C C D D C D C
and ι are equivariant for γ⊡δ. This forces γ⊡δ =γ ⊲⊳ δ. (cid:3)
D R
Theorem 4.4. If C ,C ,C are objects of C∗alg(G), then there is a unique iso-
1 2 3
morphism of triple crossed productsC ⊠(C ⊠C )∼=(C ⊠C )⊠C , which is also
1 2 3 1 2 3
G-equivariant. Thus C∗alg(G) with the tensor product ⊠ is a monoidal category.
R
Proof. An isomorphism of triple crossed products is an isomorphism that inter-
twines the embeddings of C , C and C . Since the images of these embeddings
1 2 3
generate the crossed product, such an isomorphism is unique if it exists.
Let (C ,γ ) be G-C∗-algebras and let (ϕ ,UHi) be faithful covariant represen-
i i i
tations of (C ,γ ), respectively, for i=1,2,3. The construction of the G-coaction
i i
on C ⊠C shows that (ϕ ⊠ϕ ,UHi UHj) is a faithful covariantrepresentation
i j i j
of C ⊠C on H ⊗H . Therefore, Theorem A.29 gives a faithful representation
i j i j
ϕ ⊠(ϕ ⊠ϕ ) of C ⊠(C ⊠C ) on H ⊗H ⊗H , which is characterised by:
1 2 3 1 2 3 1 2 3
ιC1(c1)7→ϕ1(c1)⊗1H2 ⊗1H3,
ιC2(c2)7→(H2⊗H3 H1) ϕ2(c2)⊗1H3 ⊗1H1 (H2⊗H3 H1)∗,
(cid:0) (cid:1)
ιC3(c3)7→(H2⊗H3 H1)(H3 H212) ϕ3(c3)⊗1H2 ⊗1H1 (H3 H212)∗(H2⊗H3 H1)∗
(cid:0) (cid:1)
for c ∈C , i=1,2,3. The diagrams in Proposition 3.4 and Corollary 3.10 give
i i
H2⊗H3 H1 =H2 H1 ·H3 H1 ,
12 23
H2⊗H3 H1·H3 H2 =H2 H1 ·H3 H1 ·H3 H2 =H3 H1⊗H2·H2 H1 .
12 12 23 12 23