Let HopfpSUqp2qq denote the free unital ˚-algebra generated by two elements a and c satisfying the following commutation relations
a˚a` c˚c“ 1 “ aa˚` q2c˚c, cc˚ “ c˚c, ac“ qca, ac˚ “ qc˚a.
(5.1)
We notice that for q “ 1 we have a commutative ˚-algebra and otherwise we have a non-commutative˚-algebra.
Consider the Hilbert space ℓ2pN0q b ℓ2pZq –i ℓ2pN0ˆ Zq with canonical orthonormal basistek,l | k P N0, l P Zu and let π0 : HopfpSUqp2qq Ñ Bpℓ2pN0q b ℓ2pZqq be defined as follows
π0paqek,l “
$&
%
αkek´1,l if ką 0 0 if k “ 0
and π0pcqek,l “ qkek,l`1. (5.2)
With a little bit of work we can show that this satisfies the relations given by 5.1 with adjoints given by
π0pa˚qek,l “ αk`1ek`1,l and π0pc˚qek,l “ qkek,l´1 (5.3)
and so this forms a˚-representation of HopfpSUqp2qq.
Notation 5.1.1 Throughout this chapter fork ă 0 we let ak “ pa˚q´k andpa˚qk “ a´k. For allk P Z and m, n P N0we denoteakmn :“ akpc˚qmcn.
The following theorem is due to Woronowicz. For a proof see Theorem 1.2 in Woronow-icz’ paper Woronowicz (1987b) or Proposition 6.2.5 in Timmermann’s book Timmer-mann(2008).
Theorem 5.1.2 The settakmn | k P Z, m, n P N0u forms a basis for HopfpSUqp2qq.
We now want to build a C˚-algebra A with HopfpSUqp2qq as a dense ˚-subalgebra. First we need to give an appropriate norm on HopfpSUqp2qq.
Lemma 5.1.3 Let π : HopfpSUqp2qq Ñ BpHq be a ˚-representation for any Hilbert space H, then we have}πpakmnq} ď 1 for all k P Z and m, n P N0.
Proof
For ξ P H we have
}πpaqξ}2` }πpcqξ}2 “ pπpaqξ|πpaqξq ` pπpcqξ|πpcqξq
“ pπpaq˚πpaqξ ` πpcq˚πpcqξ|ξq “ pπpa˚a` c˚cqξ|ξq “ pξ|ξq “ }ξ}2.
Then it follows by varying ξ P L2pSUqp2qq with }ξ} ď 1 that }πpaq} ď 1 and }πpcq} ď 1 and thus}πpakmnq} ď 1 for all k P Z and m, n P N0as π is a˚-homomorphism. ✷ Using that we have a ˚-representation π0 of HopfpSUqp2qq given by Equation (5.2) we can define a map} ¨ } : HopfpSUqp2qq Ñ R` given by
}x} “ sup
$&
%}πpxq}
ˇˇ ˇˇ ˇˇ
π : HopfpSUqp2qq Ñ BpHq for H a
Hilbert space and π a˚-representation ,.
- (5.4)
for x P HopfpSUqp2qq. We note this is finite by Lemma 5.1.3. We show in Propo-sition 5.1.5 that it is non-zero and thus a norm satisfying }x˚x} “ }x}2 for all x P HopfpSUqp2qq. We have a simple lemma first.
Lemma 5.1.4 We have fork, tP Z and m, n, s P N0 that
π0pakmnqes,t “
$’
’’
&
’’
’%
qspn`mqαs. . . αs´pk´1qes´k,t`n´m if 0 ď k ď s qspn`mqαs`1. . . αs´kes´k,t`n´m ifk ă 0
0 if k ą s
(5.5)
where ifk “ 0 this reduces to π0pa0mnqes,t “ qspn`mqes,t`n´m.
Proof
For k ą s we have π0pakqes,t “ 0. For 0 ď k ď s we have
π0pakqes,t“ αsak´1es´1,t “ αsαs´1pa˚qk´2es´2,t
“ αsαs´1αs´2pa˚qk´3es´3,t “ ¨ ¨ ¨ “ αsαs´1¨ ¨ ¨ αs´pk´1qes´k,t
and the result follows as π0ppc˚qmcnqes,t“ qspn`mqes,t`n´m. The case of k ă 0 is similar.
✷
Proposition 5.1.5 We have }akmn} ‰ 0 and }akmn} ď 1 for all k P Z and m, n P N0. In particular the completion of HopfpSUqp2qq with respect to the norm given is a C˚ -algebra.
Proof
If kě 0 let s ě k and then
}π0pakmnqes,t} “ qspn`mqαs¨ ¨ ¨ αs´pk´1q}es´k,t`n´m} “ qspn`mqαs¨ ¨ ¨ αs´pk´1q ą 0
and similarly if k ă 0 then }π0pakmnqes,t} ą 0 and so
}akmn} ě }π0pakmnq} “ sup }π0pakmnqξ} ˇ
ˇ ξ P ℓ2pN0q b ℓ2pZq, }ξ} ď 1( ą 0.
It is almost immediate from the definition that}x˚x} “ }x}2 for all x P HopfpSUqp2qq and then it follows easily that the completion is a C˚-algebra. ✷
We are now in a position to define the following C˚-algebraic completion of HopfpSUqp2qq.
In fact as we will see shortly, this gives us the C˚-algebra from the reduced C˚-algebraic quantum grouppCpSUqp2qq, ∆q.
Definition 5.1.6 We defineA to be the completion of HopfpSUqp2qq with respect to the norm given by Equation(5.4).
We can make A into a compact matrix quantum group as follows.
Theorem 5.1.7 Consideru “
¨
˝ a ´qc˚ c a˚
˛
‚ P M2pAq. Then we have a map ∆ : A Ñ AbminA given by
∆paq “ a b a ´ qc˚b c, ∆pcq “ c b a ` a˚b c (5.6)
making this a compact matrix quantum group pA, ∆, uq as per definition 3.2.22. The antipodeS as defined in Definition-Theorem2.2.7is given by
Spaq “ a˚, Spa˚q “ a, Spcq “ ´qc, Spc˚q “ ´1
qc˚. (5.7) Proof
We show that the conditions of Proposition3.2.21are satisfied. It is easy to see that u is unitary. We can also easily see that the inverse of ¯u is given by
¨
˝ a q2c
´1qc˚ a˚
˛
‚.
Clearly A “ alg tuij | 1 ď i, j ď 2u}¨} by construction where alg denotes the alge-bra generated. It is easy to show that ∆ satisfies condition (iv) in Proposition3.2.21and so we have a compact matrix quantum group. The antipode S follows from the equation Spuijq “ u˚jifor 1 ď i, j ď 2. ✷
We now calculate thepfzqzPC characters from3.2.16.
Proposition 5.1.8 Forz P C we have fzpaq “ q´z,fzpa˚q “ qzandfzpcq “ fzpc˚q “ 0.
Furthermore we have
pid b fzq∆paq “ q´za“ pfzb idq∆paq,
pid b fzq∆pa˚q “ qza˚ “ pfzb idq∆pa˚q, pid b fzq∆pcq “ q´zc, pfzb idq∆pcq “ qzc,
pid b fzq∆pc˚q “ qzc˚ and pfz b idq∆pc˚q “ q´zc˚. Proof
We calculate the F -matrix for the corepresentation u. We have by Theorem 3.2.15 that F intertwines u and S22puq and we have S22puq “
¨
˚˝
a ´1 qc˚ q2c a˚
˛
‹‚. Setting F “
¨
˝ q´1 0
0 q
˛
‚we can easily show that F u “ S22puqF and furthermore this satisfies TrpF q “ TrpF´1q giving the F -matrix for u2of Theorem3.2.15.
We can then calculate the fz values on the generators a and c and their adjoints easily (for example fzpaq “ pF11qz “ q´z). We can then use these formulas and Equation (5.5) to calculate the remainder of the equations. ✷
The formulas in the following corollary can be extended to all HopfpSUqp2qq as R is a
˚-anti-homomorphism and τzis a homomorphism on HopfpSUqp2qq.
Corollary 5.1.9 Let z P C. We have the following formulas for the scaling group on HopfpSUqp2qq:
τzpaq “ a, τzpcq “ q2izc, τzpc˚q “ q´2izc˚, τzpa˚q “ a˚
and for the unitary antipodeR on HopfpSUqp2qq we have
Rpaq “ a˚, Rpcq “
$&
%
´c if 0 ă q ď 1 c if ´ 1 ď q ă 0.
Proof
The first set of formulas for τz follow easily from Proposition 3.2.18 and the formulas in Proposition 5.1.8. It is easy to calculate R knowing S and τ´i{2 on the Hopf algebra elements. It follows that R is a˚-anti-homomorphism from Definition-Theorem2.2.7and that τz is a homomorphism from Proposition1.3.10. ✷
Finally we have the following important theorem. We refer the reader to B´edos et al.
(2001) for a proof.
Theorem 5.1.10 The compact quantum group SUqp2q is co-amenable.
As SUqp2q is coamenable we have that CpSUqp2qq –i A from Theorem3.4.1and so we can always work with the reduced C˚-algebraic quantum group CpSUqp2qq Ă BpL2pSUqp2qqq.