Woronowicz construction of compact quantum groups for functions on permutations. Classification result for N=3

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by  Anna Kula

WORONOWICZ CONSTRUCTION OF COMPACT QUANTUM
GROUPS FOR FUNCTIONS ON PERMUTATIONS.
CLASSIFICATION RESULT FOR N = 3
ANNA KULA
Abstract. We provide a classification of compact quantum groups, which can be ob-
tainedbytheWoronowiczconstruction,whenthearraysusedinthetwisteddeterminant
4 condition are extensions of functions on permutations. Generalproperties of suchquan-
1 tum groups are revealed with the aid of operators intertwining tensor powers of the
0 fundamental corepresentation. Two new families of quantum groups appear: three ver-
2
sions of the quantum group SU (3) with complex q and a non-commutative analogue of
q
n the semi-direct product of two-dimensional torus with the alternating group A3.
a
J

8
2

]
A 1. Introduction
O
The Woronowicz construction [Wor88] provides a very general and elegant method of
.
h creating compact quantum groups. According to this construction, the universal unital
at C∗-algebra generated by abstract elements making up a N × N unitary matrix u and
m satisfying a modified (twisted) determinant condition admits the quantum group struc-
[ ture. The modification of the determinant condition depends on a choice of NN complex

1 constants Ei1,i2,...,iN, ik ∈ {1,2,...,N}, forming a ’nondegenerate array’. The nondegen-
v eracy condition still leaves a lot of freedom to choose the constants–at least in theory. In
3
practice, it is extremely difficult to find an example which is non-trivial. The latter means
4
1 that the C∗-algebra is noncommutative and thus the quantum group does not originate
7
from any classical group.
.
1
Until present there have been only few examples of quantum groups successfully con-
0
4 structed with the aid of the Woronowicz result. These are: the twisted special unitary
1 group SU (N) [Wor88], the q-deformed unitary group U (2) [Wys04] and the q-deformed
q q
:
v special orthogonal group SO (3) [Lan98]. Recently, the author showed that also a twisted
q
Xi product of SU−1(2) with the two-dimensional torus can be added to this list (see [Kul13]).
r On the other hand, the construction is fruitful in the sense that it helped describing
a
the quantum group structure. For instance, Wysoczański observed that U (2) is a twisted
q
product of SU (2) with the unit circle, while Lance found a C∗-embedding of SO (3)
q q
into SU (2). Both results allowed to provide systematic descriptions of irreducible
q1/2
representations of the quantum groups in question as well as the related Haar states.
Several examples of objects coming from Woronowicz construction, which are known,
contrast vividly with the broad range of choice of the constants in the array E =
[E ]N . It is thus natural to wonder what kind of arrays E lead–through
i1,i2,...,iN i1,i2,...,iN=1
the Woronowicz construction–to non-trivial quantum groups. However, different arrays
Date: January 29, 2014.
2010 Mathematics Subject Classification. Primary: 81R50, Secondary: 46L65, 18D10.
Key words and phrases. compact quantum groups, twisted determinant condition, intertwining
operators.
1
2 ANNAKULA
can lead to the same quantum group (as will become clear from the results of this pa-
per); therefore, it is better to rephrase the question as follows: what are the non-trivial
quantum groups which can be obtained from the Woronowicz construction?
One can observe that all the examples, except SO (3), are related to arrays which
q
are trivial extensions of nowhere-zero functions on permutations. By this we mean that
E is non-zero if and only if (i ,i ,...,i ) is a permutation of the set {1,2,...,N}.
i1,i2,...,iN 1 2 N
For instance, for SU (N) the non-zero constants are given by E = (−q)inv(σ), where inv
q σ
denotes the function counting the number of inversion in the permutation. In case of
the quantum group U (2), the function used in the construction associates a permutation
q
σ ∈ S with the value (−q)3−c(σ), where c(σ) stands for the number of cycles in σ. In the
3
last known case of SU (2)× T2, a transposition-coloring function was used (see [Kul13]
−1 α
for details).
TheaimofthispaperistostudycompactquantumgroupscomingfromtheWoronowicz
construction under the assumption that arrays E are trivial extensions of (nowhere-zero)
functions on permutations. In such case, the nondegeneracy condition is always satisfied.
Moreover, we can use the idea included in the proof of the Woronowicz construction to
derive modularproperties of thequantum groupaswell ascommutation relationsbetween
its generators. In particular, we show (Corollary3.4) that such a quantum groupis always
a quantum subgroup of the van Daele and Wang’s universal unitary quantum group
A (M), whereM isadiagonalmatrixwithcoefficients depending onE. Morphisms which
u
intertwine tensor powers of the fundamental corepresentation are the key ingredients in
these calculations. This methodreplaces routinebut tedious calculations (usually omitted
as in [Bra89] or [Lan98]) and, to the best of our knowledge, it has not yet been used in
this context.
The main result of this paper (Theorem 2.3) asserts that the Woronowicz construction
for N = 3, with E as above, allows only a restricted type of compact quantum groups.
Some of them areknown: the quantum version of the two-dimensional torus, the quantum
U (2) with complex q [ZZ05], the quantum SU (3) with q real. Nevertheless, two new
q q
families appear too; these are three versions of SU (3) with complex parameter q and a
q
non-commutative analogue of the semi-direct product of two-dimensional torus T2 with
the alternating group A , denoted by A (3). We will focus primarily on the question
3 p,k,m
of non-triviality of the new quantum groups, postponing deeper studies of their structure
to a separate paper.
The classification result may be interpreted as reflecting a very specific way in which
the construction was carried out. In fact, the twisted determinant condition comes from
the property that the third power of the fundamental corepresentation intertwines the
identity and, in this sense, it mimics the property of the classical special unitary group.
But the classification result reveals also a great intuition of S. L. Woronowicz who was
able to find a rare set of parameters leading to the non-trivial quantum group SU (3).
q
Let us mention that the analogous classification problem for N = 2 is already solved.
Indeed, usingTheoremA1.1from[Wor87a],onecandeducethatifanarrayE = [E ]
ij i,j=1,2
is a trivial extension of a function on permutations, then the only quantum groups which
we get by the Woronowicz construction are the twisted SU (2) (if E21 = −q is real) and
the quantum version of the unit circle T (otherwise). q E12
The paper is organized as follows. In Section 2 we recall the Woronowicz construction
and gather necessary definitions. Modular properties as well as commutation relations
between generatorsarededuced frommorphismproperties inSections 3and4. Additional
relations, which do not refer explicitly to morphism property, are presented in Section
WORONOWICZ CONSTRUCTION FOR N =3 3
5. In Section 6 we define the new quantum groups arising from the construction and
answer the question of their non-triviality. Section 7 is devoted to study the case when
the fundamental corepresentation decomposes into two blocks. Finally, the proof of the
classification theorem represents the content of Section 8.
Throughout the paper, we assume that N = 3, although the general theory recalled
in Section 2 can be stated for arbitrary dimension. We use the notation C∗ = C \ {0},
R∗ = R\{0}. A scalar product will always be linear with respect to the second variable.
2. General theory
We recall in this Section the construction from [Wor88] and some notation to be used
in the sequel. We refer to [Wor87b] and [Tim08] for more detailed information about
compact quantum groups.
There exist various (not always equivalent) definitions of compact quantum groups.
The one which we adopt here comes from [Wor87b], see also [vDW96], and is the most
suitable in our context. By a compact (matrix) quantum group we understand a pair
G = (A,u), where A is a unital C∗-algebra and u is a N ×N matrix with coefficients in
A such that:
(1) the coefficients u , i,j = 1,...,n, generate A,
ij
(2) thereexists aC∗-homomorphism∆ : A → A⊗A such that∆(u ) = n u ⊗u ,
ij k=1 ik kj
(3) the matrices u = [u ] and ut = [u ] are invertible.
ij ji
P
The∗-subalgebrageneratedbyu ’sadmitstheHopfstructure, withthecounitε(u ) = δ
ij ij ij
and the antipode κ(u ) = (u−1) .
ij ij
ForafixedN ∈ NletusconsideraNN-arrayofcomplexnumbersE = [E ]N .
i1,i2,...,iN i1,i2,...,iN=1
We say that the array E is left nondegenerate if the arrays E ,...,E are linearly
1− N−
independent, where E = [E ]N . Similarly, the array E is said to be
k− k,i2,...,iN i2,...,iN=1
right nondegenerate if the arrays E ,...,E are linearly independent, where E =
−1 −N −k
[E ]N .
i1,...,iN−1,k i1,...,iN−1=1
Theorem 2.1 (Woronowicz Construction, Theorem 1.4 in [Wor88]). Let E be a left and
right nondegenerate NN-array of complex numbers, and let A be the universal C∗-algebra
with unit 1A generated by matrix coefficients u = [ujk]Nj,k=1 satisfying
a) unitarity condition:
N N
(U) ujsu∗ks = δjk1A = u∗sjusk
s=1 s=1
X X
b) twisted determinant condition:
N
(TD) Ei1,i2,...,iNua1i1ua2i2...uaNiN = Ea1,a2,...,aN ·1A.
i1,i2X,...,iN=1
Then the pair G = (A,u) is a compact quantum group.
We shall call G as above a compact quantum group related to E, coming from Woronow-
icz construction. The quantum group structure on G is then imposed by the fact that
u is a corepresentation, so the usual formulas for comultiplication, counit and antipode
hold. Namely,
N
∆(u ) = u ⊗u , ε(u ) = δ , κ(u ) = u∗ .
jk js sk jk jk jk jk
s=1
X
4 ANNAKULA
The matrix u is called the fundamental corepresentation.
Given a compact quantum group G there exists the Haar state h ∈ A′ and it is a KMS-
state, i.e. h(ab) = h(bσ (a)) for any a,b ∈ A, where σ is an analytic extension of the
−i −i
modular automorphisms group (σt)t∈R (see for example [BR97, Definition 5.3.1]). The
KMS-property encodes the modular properties of G, which describe how far from being
tracial the Haar state is. If h happens to be tracial, then we say that G is of Kac type.
It is well known (cf. [Wor87b]) that if the C∗-algebra A of a compact quantum group
is commutative, then there exists a (classical) compact group G ⊂ GL(N) such that
A = C(G) is the algebra of all continuous functions on G, the generator u associates
ij
a matrix g ∈ G with the (i,j)-th coefficient of g, and the comultiplication of functions
from A reflects the group multiplication: ∆(f)(g,g′) = f(gg′). Then G = (A,[u ]N ) is
ij i,j=1
usually called a quantum version of the group G. Quantum versions of classical groups
represent a very well-understood class of compact quantum groups (in particular, they
are always of Kac type). That is why, in this paper, we shall refer to them as to trivial
compact quantum groups.
We say that a NN-array E satisfies the permutation condition if
(P) E 6= 0 if and only if (i ,i ,...,i ) ∈ S .
i1,i2,...,iN 1 2 N N
This means that there exists a complex, nowhere-zero function f on S , the set of per-
N
mutations of N elements, such that E is the trivial extension of f, that is
f(σ) if (i ,i ,...,i ) = σ ∈ S
E = 1 2 N N .
i1,i2,...,iN 0 if (i ,i ,...,i ) 6∈ S
1 2 N N
(cid:26)
It is easily seen that the permutation condition (P) ensures that the array E is left and
right nondegenerate and thus the Woronowicz construction applies.
It will be convenient to use the following (technical) definition.
Definition 2.2. A PW-quantum group of dimension N is a compact quantum group G
for which there exists a N-dimensional array E satisfying the permutation condition (P)
and such that G comes from the Woronowicz construction related to the array E.
Let E be an array related to a PW-quantum group (in particular, E must satisfy the
permutation condition). Then we can multiply all constants by a non-zero scalar and
the relations remain unchanged. Therefore, without lose of generality, we can assume the
normalization E = 1.
12...N
The main problem investigated in this paper can now be shortly rephrased: classify all
PW-quantum groups of dimension 3. The answer which we aim to show is the following.
Theorem 2.3 (Classification Theorem). The following is a complete list of PW-quantum
groups of dimension 3:
(1) C(T2), quantum version of two-dimensional torus,
(2) U (2) for q ∈ C∗, see Definition 6.1,
q
(3) A (3) for p ∈ C∗ and k,m ∈ {0,1,2}, see Definition 6.4,
p,k,m
(4) SU (3) for p ∈ C∗ and m ∈ {0,1,2}, see Definition 6.2.
p,m
The proof of Theorem will be given in Section 8. We emphasize, however, that the
meaning of this result is that, although a lot of freedom is given in the Woronowicz
construction, only few quantum groups can appear. Moreover, it will become clear from
Section 6 that the non-trivial ones are U (2) with q 6= 1, SU (3) with p 6= 1 and m 6= 0,
q p,m
and A (3) with k 6= m and k +m 6∈ 3N.
p,k,m
WORONOWICZ CONSTRUCTION FOR N =3 5
Remark 2.4. For N = 3, an array E satisfying the permutation condition (P) and the
normalization E = 1 is uniquely defined by 5 complex, non-zero parameters. In the
123
sequel, we shall frequently use these parameters, adopting the following notation:
E = 1, E = p, E = q,
123 132 213
E = r, E = s, E = t.
231 312 321
Remark 2.5. The indices used to number the generators in the array u = [u ]3 always
ij i,j=1
change in {1,2,3}. We try to use r,i,k to number the rows in the array and n,j,l
to number the columns. This is not always possible due to the antipode property, see
Corollary 3.4. Moreover, starting from Section 4.2, whenever a triple (r,i,k) or (n,j,l)
appear, we tacitly assume that this is an element of S . For example, when r and i are
3
given and r 6= i, then the index k is the unique element different from the other two, that
is such that (r,i,k) ∈ S .
3
3. Morphisms of PW-quantum groups
Throughout this Section, whenever G appears, it is understood to be a PW-quantum
group related to a normalized array E of dimension 3. We shall deduce here several
relations which hold for generators of such G. For that purpose we recall the main idea of
the proof of the Woronowicz construction, which establishes morphisms between tensor
powers of the fundamental corepresentation. Then we show how the morphism properties
translate into relations.
3.1. Idea of the Woronowicz construction. Eachcompactquantum groupG = (A,u)
gives rise to the so-called complete concrete monoidal W∗-category R, cf. [Wor88]. In this
category, objects Obj(R) are finite-dimensional unitary corepresentations u on (some)
s
Hilbert spaces H with a distinguish element u–the fundamental corepresentation. Mor-
s
phisms of this category are intertwining operators Mor(u ,u ) := {T ∈ B(H ,H ) :
s t s t
u (T ⊗id) = (T ⊗id)u }, where id denotes the identity on A.
s t
In [Wor88, Theorem 1.3], Woronowicz showed that the converse is true. Namely, if R
is a complete concrete monoidal W∗-category with a distinguished object u, and if the
conjugate object u¯ exists and {u,u¯} generates Obj(R), then the universal C∗-algebra A
generated by matrix coefficients of u, together with u as fundamental corepresentation is
a compact quantum group.
This result lies in the core of the proof of the Woronowicz construction. Namely, given
the C∗-algebra A with u satisfying (U) and (TD), it is possible to construct a complete
concrete monoidal W∗-category R(G), which leads to G = (A,u). Objects of this category
are generated by tensor powers of the matrix u
Obj (G) := {u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)n : n ∈ N}.
0
Morphisms are linear combinations of elements which are composition of mappings of the
form I ⊗E ⊗I and I ⊗E∗⊗I , where E is a fixed operator. The operator E serves as
k l k l
the main building block for morphisms between different elements in Obj(G).
3.2. Morphisms. Let e ,e ,e be the standard orthonormal basis in C3. The funda-
1 2 3
mental corepresentation u belongs to M (A) ∼= B(C3) ⊗ A and thus can be expressed
3
as
u = m ⊗u ,
kl kl
k,l
X
6 ANNAKULA
where mkl are standard matrix units (i.e. mklej = δjkek). Given two elements v ∈ Mn′(A),
w ∈ Mn′′(A), for some n′,n′′ ∈ N,
n′ n′′
v = m′ ⊗v w , w = m′′ ⊗v w ,
jk jk st st jk st
j,k=1 s,t=1
X X
where m′jk and m′s′t are matrix units in Mn′(A) and Mn′′(A) respectively, we define the
tensor product of corepresentations v and w
n′ n′′
v(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)w = m′jk ⊗m′s′t ⊗vjkwst.
j,k=1s,t=1
X X
LetH = (C3)⊗n andletI denotestheidentityoperatoronH . Wedefinethefollowing
n n n
mappings:
E : C ∋ 1 7→ E e ⊗e ⊗e ∈ H ,
ijk i j k 3
i,j,k
X
P = (E∗ ⊗I )(I ⊗E) : H ∋ e 7→ E¯ E e ∈ H ,
1 1 1 x xjk jka a 1
!
a j,k
X X
Q = (E∗ ⊗I )(I ⊗E) : H ∋ e ⊗e 7→ E¯ E e ⊗e ∈ H .
2 2 2 x y xyk kab a b 2
!
a,b k
X X
The explicit expressions follow from E∗(e ⊗e ⊗e ) = E¯ . Moreover, we observe that
i j k ijk
the space H¯ := Q(H ) is spanned by the vectors
2
x = E e ⊗e , k = 1,2,3
k kab a b
a,b
X
(cf. formula (4.2) in [Wor88]). Under the permutation assumption for E, the vectors
x ,x ,x form an orthogonal basis (not orthonormal in general).
1 2 3
Furthermore, let R denotes the embedding of H¯ into H . Define morphisms
2
T := (I ⊗R∗)E : C ∋ 1 7→ e ⊗x ∈ H ⊗H¯,
1 i i 1
i
X
T¯ : H¯ ⊗H ∋ x ⊗e 7→ δ ∈ C.
1 j k jk
and let
3
s = m¯ ⊗u∗ ∈ M (A) ∼= B(H¯)⊗A,
st st 3
s,t=1
X
where m¯ (j,k = 1,2,3) are the matrix units in B(H¯) with respect to the orthogonal
jk
basis x ,x ,x , i.e. m¯ x = δ x .
1 2 3 jk n kn j
It follows from (U) and (TP) (see the proof of Theorem 1.4 in [Wor88]) that
(a) s is the corepresentation conjugate to u (denoted in the sequel by u¯),
(b) E ∈ Mor(1,u(cid:31)(cid:24)⊤(cid:30)(cid:25)(cid:29)(cid:26)(cid:28)(cid:27)3),
(c) P ∈ Mor(u,u), Q ∈ Mor(u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2,u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2), R ∈ Mor(u¯,u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2),
(d) T ∈ Mor(1,u(cid:31)(cid:24)⊤(cid:30)(cid:25)(cid:29)(cid:26)(cid:28)(cid:27)u¯), T¯ ∈ Mor(u¯(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)u,1).
Note that (b) encodes the twisted determinant condition (TP) whereas the condition (U)
is hidden in (d).
In the following we shall translate these morphism properties into relations between
algebra generators u ’s.
jk
WORONOWICZ CONSTRUCTION FOR N =3 7
3.3. Decomposition of u. ForG related to thearray E we define the diagonal constants
(1) p = E¯ E for j = 1,2,3.
j jab abj
a,b
X
Lemma 3.1. If p 6= p then u = u = 0.
j k kj jk
Proof. The operator P = (E∗ ⊗I )(I ⊗E) is by definition a morphism between u and
1 1
itself, and (as mentioned above) Pe = E¯ E e . But since E 6= 0
j k a,b jab abk k jln
iff (j,l,n) ∈ S , the only non-zero coefficient(cid:16)on the right-(cid:17)hand side will be the one
3 P P
standing by e . Thus P is diagonal, Pe = p e . The fact that P ∈ Mor(u,u), i.e.
j j j j
u(P ⊗id) = (P ⊗id)u means that the two terms
u(P ⊗id) = mkl ⊗ukl mjj ⊗pj1A = mkj ⊗pjukj
k,l j j,k
(cid:0)X (cid:1)(cid:0)X (cid:1) X
(P ⊗id)u = mjj ⊗pj1A mkl ⊗ukl = mjl ⊗pjujl
j k,l j,l
(cid:0)X (cid:1)(cid:0)X (cid:1) X
are equal (above, we use m m = δ m ). Comparing the corresponding terms, we see
kl ij il kj
that p u = p u , from which the conclusion follows. (cid:3)
j kj k kj
Theorem 3.2 (Decomposition of the fundamental representation). Let G be a PW-
quantum group related to E, with diagonal constants p ,p ,p . Then unless p = p = p ,
1 2 3 1 2 3
u is reducible. More precisely, if the diagonal constants are pairwisely different, then
u = u ⊕ u ⊕ u . If p = p 6= p then u decomposes into two blocks of dimension 1
11 22 33 i k r
and 2.
Note that the fact that p = p = p does not imply that u is irreducible.
1 2 3
3.4. Modular properties. Let us define the modular constants related to the array E
by
(2) M = |E |2 for n = 1,2,3.
n njk
jk
X
These constants encode the modular properties of the related PW-quantum group in the
sense which will be clear from Corollary 3.4. First, we show that the transpose of u is
similar to a unitary matrix.
Theorem 3.3 (Modular properties). Let G be a PW-quantum group related to E, with
modular constants M ,M ,M . Then the generators of G satisfy
1 2 3
3 3
1 1
(3) Msusau∗sb = δabMa1A, M u∗kiusi = M δks1A.
i k
s=1 i=1
X X
Proof. Recall that T is a linear operator from C to H ⊗H¯ defined by T(1) = e ⊗x ,
1 i i i
and is a morphism between 1 and u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)u¯. Then from the condition CMW∗ III in [Wor88]
P
it follows that T∗ ∈ Mor(u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)u¯,1). It can be checked directly that
T∗(e ⊗x ) = δ M , where M = kx k2.
a b ab a a a
Thus (T∗ ⊗id)(ea ⊗xb ⊗1A) = δabMa1A while
(T∗ ⊗id)(u(cid:31)(cid:24)⊤(cid:30)(cid:25)(cid:29)(cid:26)(cid:28)(cid:27)u¯)(ea ⊗xb ⊗1A) = (T∗ ⊗id) mijea ⊗m¯stxb ⊗uiju∗st
i,j,s,t
(cid:0) X (cid:1)
= T∗(e ⊗x )⊗u u∗ = M u u∗ .
i s ia sb s sa sb
i,s s
X X
8 ANNAKULA
So (T∗ ⊗id)(u(cid:31)(cid:24)⊤(cid:30)(cid:25)(cid:29)(cid:26)(cid:28)(cid:27)u¯) = (T∗ ⊗id) yields the first relation in (3).
Toshowthesecondrelation,weusetheoperatorT¯ : H¯⊗H → CdefinedbyT¯(x ⊗e ) =
1 j k
δ . Its adjoint is
jk
1
T¯∗(1) = x ⊗e , M = kx k2.
i i i i
M
i
i
X
Since we know that T¯ ∈ Mor(u¯(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)u,1), thus T¯∗ ∈ Mor(1,u¯(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)u). Evaluating the morphism
condition on 1⊗1A we have
1
xi ⊗ei ⊗ 1A = (T¯∗ ⊗id)(1⊗1A) = (u¯(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)u)(T¯∗⊗id)(1⊗1A)
M
i
i
X
1 1
= m¯kj ⊗mst ⊗u∗kjust ( M xi ⊗ei ⊗1A) = xk ⊗es ⊗ M u∗kiusi.
i i
k,j,s,t i k,s i
(cid:0) X (cid:1) X X X
Comparing the coefficients we find out that the second relation in (3) holds. To end the
proof, it remains to see that kx k2 = |E |2. (cid:3)
n jk njk
Note that the relations (3) can be rPead as
utMu¯M−1 = I = Mu¯M−1ut,
where M = diag(M ,M ,M ). This fits with the definition of the universal unitary
1 2 3
quantum group, cf. [vDW96]. The next result combine this observation with the standard
results from [Wor87b], see also [Tim08].
Corollary 3.4. Let G be a PW-quantum group related to E, with modular constants
M ,M ,M . Then G is a quantum subgroup of the universal unitary quantum group
1 2 3
A (M), where M = diag (M ,M ,M ). Moreover,
u 1 2 3
(1) the antipode of G is uniquely defined by
M
(4) κ(u ) = u∗ and κ(u∗ ) = ju ;
kj jk kj M jk
k
(2) the Woronowicz characters constants of G are given by
1
f (u ) = λM δ with λ = ( )( M )−1;
z ij j ij i
M
r i
X X
(3) if all M ’s are equal, then G is of Kac type.
j
3.5. Adjoints of generators. The morphism R will help us recover the formula for
adjoints of the generators. Let us recall (cf. Remark 2.5) that for r ∈ {1,2,3} we can find
i,k ∈ {1,2,3} such that i 6= k and (r,i,k) ∈ S .
3
Theorem 3.5. Given r,n ∈ {1,2,3}, the adjoint of the generator u in G satisfies
rn
E E
(5) u∗ = njlu u + nlju u
rn E ij kl E il kj
rik rik
and
E¯ E¯
(6) u∗ = riku u + rkiu u ,
rn E¯ ij kl E¯ kj il
njl njl
where i,k and j,l are chosen to complete r and n, respectively, to a permutation.
WORONOWICZ CONSTRUCTION FOR N =3 9
Proof. Recall that R is the embedding of H¯ into H , where
2
3
H¯ = { α x ;α ∈ C}, x = E e ⊗e .
k k k k kab a b
k=1 a,b
X X
This means that
Rx = E e ⊗e ∈ H .
k kab a b 2
a,b
X
Using u¯ = m¯ ⊗ u∗ , we compare both sides of (R ⊗ id)u¯ = u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2(R ⊗ id) when
st st
s,t
X
evaluated on xk ⊗1A:
(R⊗id)u¯(xk ⊗1A) = Rm¯stxk ⊗u∗st = ea ⊗eb ⊗ Esabu∗sk
!
s,t a,b s
X X X
and
u(cid:31)(cid:24)⊤(cid:30)(cid:25)(cid:29)(cid:26)(cid:28)(cid:27)2(R⊗id)(xk ⊗1A) = mij ⊗mst ⊗uijust Ekabea ⊗eb ⊗1A
i,j,s,t a,b
(cid:0) X (cid:1)(cid:0)X (cid:1)
= E m e ⊗m e ⊗u u = e ⊗e ⊗ E u u .
kab ij a st b ij st i s kjt ij st
!
i,j,s,t a,b i,s j,t
X X X X
Hence
E u∗ = E u u .
sib sk kjt ij bt
s j,t
X X
Due to the permutation condition (P), for fixed i 6= b there exists the unique s such that
(i,b,s) ∈ S and E 6= 0, and exactly two pairs of (j,t) such that (k,j,t) ∈ S , therefore
3 sib 3
E E
u∗ = kjtu u + ktju u .
sk E ij bt E it bj
sib sib
Changing indices appropriately, we get (5).
PerformingsimilarcomputationsforR∗(e ⊗e ) = E¯ x ,whichisamorphismsbetween
j l njl n
u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2 and u¯, we end up with the relation (6). (cid:3)
4. Commutation relations for generators
We devote this section to exhibit the commutation relations between generators. For
that we use the fact that Q = (E∗⊗I )(I ⊗E) and its adjoint, acting on H , intertwine
2 2 2
u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2 with itself. It turns out that the commutation relations depend on the constants
c(n), defined, for l 6= n, as
l
E E
(7) c(n) := jln nlj,
l E E
jnl lnj
where j is the unique integer such that (j,l,n) ∈ S . We shall refer to c(n)’s as to char-
3 l
acteristic constants of the array E. There are six such constants, but it is immediately
seen that they satisfy
(8) c(j)c(n) = 1, c(l)c(j) = c(j).
n j n l n
The main result of this Section is the following.
Theorem 4.1. Let l 6= n, then the generators in G satisfy the following commutations:
10 ANNAKULA
(a) If c(n) 6= 1 then
l
(9) u u = 0 and u u = 0 (a = 1,2,3)
an al la na
and for r 6= k we have
E E 1−c(r) E 1−c(r)c(n)
(10) u u = irk jln k u u + irk k l u u .
rn kl Eikr Ejnl1−c(n) kl rn Eikr 1−c(n) kn rl
l l
(b) If c(n) = 1 then
l
E E¯
jln jln
(11) u u = − u u , u u = − u u
an al E al an na la E¯ la na
jnl jnl
and
E E E E −1
(12) u u = irk jln 1+c(r) ikr 2 1+ jln 2 u u
rn kl E E k E E kl rn
ikr jnl (cid:18) irk (cid:19)(cid:18) jnl (cid:19)
+ Eirk 1−c(r) E(cid:12)(cid:12)jln 2 (cid:12)(cid:12)Eikr 2 (cid:12)(cid:12)1+ (cid:12)(cid:12)Ejln 2 −1 u u .
E k E E E kn rl
ikr (cid:18) jnl irk (cid:19)(cid:18) jnl (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Corollary 4.2. If k 6= r and l 6= n(cid:12), then(cid:12) t(cid:12)here(cid:12)exist com(cid:12)plex c(cid:12)onstants An,l and Bn,l
r,k r,k
such that
u u = An,lu u +Bn,lu u .
rn kl r,k kl rn r,k kn rl
The rest of this Section is devoted to the proof of Theorem 4.1.
4.1. Intertwiner Q. We consider the operator Q = (E∗ ⊗ I )(I ⊗ E) : H → H and
2 2 2 2
write it as
Q = Q m ⊗m , where Q = he ⊗e ,Q(e ⊗e )i = E E¯ .
ab,xy ax by ab,xy a b x y iab xyi
a,b,x,y i
X X
Then, with u = m ⊗ u , the fact that Q ∈ Mor(u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2,u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2), i.e. (Q ⊗ I)u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2 =
l,n ln ln
u(cid:24)(cid:31)⊤(cid:25)(cid:30)(cid:26)(cid:29)(cid:27)(cid:28)2(Q⊗I), means explicitly that
P
[Q·(mln ⊗ml′n′)]⊗ulnul′n′ = [(mln ⊗ml′n′)·Q]⊗ulnul′n′.
l,n,l′,n′ l,n,l′,n′
X X
Application of m m = δ m to express
ab cd b,c ad
Q·(mln ⊗ml′n′) = Qab,xy(maxmln ⊗mbyml′n′) = Qab,ll′man ⊗mbn′,
a,b,x,y a,b
X X
(mln ⊗ml′n′)·Q = Qab,xy(mlnmax ⊗ml′n′mby) = Qnn′,xymlx ⊗ml′y,
a,b,x,y x,y
X X
leads to
man ⊗mbn′ ⊗ Qab,ll′ulnul′n′ = mlx ⊗ml′y ⊗ Qnn′,xyulnul′n′
a,n,b,n′ l,l′ l,x,l′,y n,n′
X X X X
Comparing the coefficients of the same matrix units we find out that for fixed A,B,X,Y
(13) QAB,ll′ulXul′Y = Qnn′,XY uAnuBn′
l,l′ n,n′
X X