Some more technical basics
1
Free product of compact quantum groups
Let Q1 , Q2 be two compact quantum groups with respective coproduct
∆1 , ∆2 . Let Q = Q1 ∗ Q2 be their free product. We want to give it
a compact quantum group structure. Suppose that j1 , j2 be the embeddings of Q1 , Q2 into Q respectively. Consider the unital ∗-homomorphisms
πk = (jk ⊗ jk ) ◦ ∆k : Qk → Q ⊗ Q, for k = 1, 2. By the universal property
of the free product, we get a unique ∗-homomorphism ∆ : Q → Q ⊗ Q such
that ∆ ◦ jk = πk for k = 1, 2. It is easy to verify that ∆ is a coproduct on Q
which makes it into compact quantum group. This construction can easily
be extended to an arbitrary family of compact quantum groups, to equip
∗i∈I Qi with a coproduct which makes it a compact quantum group.
Given unitary representations Ui of Qi , i ∈ I, on Hilbert spaces Hi
(say), we can construct a ‘free product representation’ in the following way.
condier
H to be the direct sum of the Hi ’s and define U to be the direct sum
L
j
(U
i ), where ji is the canonical embeddings. It is then easy to verify
i i
that U is a unitary representation of ∗i Qi in H.
2
Quantum subgroups, Woronowicz C ∗ -ideals and
quotients
Let (Q, ∆) be a compact quantum group (CQG).
Definition 2.1 A Woronowicz subalgebra of Q is a C ∗ -subalgebra Q1 of Q
such that ∆(Q1 ) ⊆ Q1 ⊗ Q1 .
Remark 2.2 For a compact group G and a closed normal subgroup H,
C(G/H) ≡ {f ∈ C(G) : f (gh) = f (g) ∀g ∈ G, h ∈ H} is a Woronowicz subalgebra of C(G). Thus, Woronowicz subalgebras are generalizations
of quotient by normal subgroups.
Definition 2.3 A Woronowicz C ∗ -ideal is a closed two sided ideal I of Q
such that ∆ maps I to the closed linear span of Q ⊗ I + I ⊗ Q.
The importance of this concept is that the coproduct ∆ descends to a ∗˜ I (a)) := (πI ⊗ πI ) ◦ ∆(a) (where πI : Q → Q/I is the
homomorphism ∆(π
quotient map), for a ∈ Q from the quotient C ∗ algebra Q/I to (Q/I) ⊗
(Q/I), making Q/I into a CQG.
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Definition 2.4 A morphism of compact quantum group from a CQG (Q1 , ∆1 )
to another CQG (Q2 , ∆2 ) is a unital ∗-homomorphism π : Q1 → Q2 satisfying ∆2 ◦ π = (π ⊗ π) ◦ ∆1 . It is called isomorphism of CQG if it is an
isomorphism at the level of C ∗ algebras.
It is easy to verify that kernels of morphisms of CQG are Woronowicz C ∗
ideals.
Definition 2.5 A CQG (S, Φ) is called a quantum subgroup of (Q, ∆) if
there is a surjective morphism from (Q, ∆) to some CQG (S 0 , Φ0 ) which is
isomorphic as a CQG to (S, Φ).
Remark 2.6 If Q = C(G) for a compact group G, then the Woronowicz
ideals are in 1-1 correspondence with the closed subgroups of G, given by
H ↔ IH , where IH = {f ∈ C(G) : f |H ≡ 0}. Thus, Q/IH ∼
= C(H).
3
Action of compact quantum groups
We can generalize the notion of group action on spaces to the setting of
quantum groups. Given a compact group G acting on a compact Hausdorff
space X, one can define the ∗-homomorphism from C(X) to C(X)⊗C(G) ∼
=
C(X × G) by sending f ∈ C(X) to β(f )(x, g) = f (gx). This automatically
extends to the following:
Definition 3.1 We say that a CQG (Q, ∆) (co)-acts on a unital C ∗ -algebra
B if there is a unital ∗-homomorphism (called an action) α : B → B ⊗ B ⊗ Q
satisfying the following: 1. (α ⊗ id) ◦ α = (id ⊗ ∆) ◦ α,
2. the linear span of α(B)(1 ⊗ Q) is dense in B ⊗ Q.
It has been shown by Podles that (2) implies to the existence of a dense
∗-subalgebra B0 of B such that α maps B0 into the algebraic tensor product
of B0 and the canonical Hopf ∗-algebra Q0 of Q, and moreover, one has
(id ⊗ ) ◦ α = id on B0 , where is the counit of Q0 .
Definition 3.2 We say that an action α is faithful, if there is no proper
Woronowicz subalgebra Q1 of Q such that α(B) ⊆ B ⊗ Q1 .
For a CQG (Q, ∆), denote by IrrQ , the set of inequivalent, unitary irrereducible representations of Q and let uγ be a representation of Q of dimension
dγ , for γ ∈ IrrQ . We will call a vector subspace V ⊆ B a subspace correponding to uγ if
• dimV = dγ ,
2
• α(ei ) =
Pdγ
k=1 ek
d
γ
⊗ uγki , for some orthonormal basis {ej }j=1
of V.
Proposition 3.3 Let α be an action of a CQG (Q, ∆) on a C ∗ -algebra B.
Then there exist vector subspaces {Wγ }γ∈IrrQ (called spectral subspaces) of
B such that
1. B = ⊕γ∈IrrQ Wγ
2. For each γ ∈ IrrQ , there exist a set Iγ and vector subspaces Wγi , i ∈
Iγ , such that
a. Wγ = ⊕i∈Iγ Wγi .
b. Wγi corresponds to uγ for each i ∈ Iγ .
3. Each vector subspace V ⊆ B corresponding to uγ is contained in Wγ .
4. The cardinal number of Iγ doesn’t depend on the choice of {Wγi }i∈Iγ .
It is denoted by cγ and called the multiplicity of uγ in the spectrum of
α.
Definition 3.4 Suppose a CQG (Q, ∆) acts on a C ∗ -algebra B. Then B is
called
1. A quotient of (Q, ∆) by a quantum subgroup (S, ∆|S ) if:
a) B is C ∗ -isomorphic to the algebra C := {x ∈ Q : (π ⊗ id)∆(x) =
1 ⊗ x},
b) the coaction α is given by α := ∆|C ,
where π is the CQG morphism from Q to S.
2. Embeddable, if there exists a faithful C ∗ -homomorphism ψ : B → Q
such that
∆ ◦ ψ = (ψ ⊗ id) ◦ α.
3. Homogeneous if the multiplicity of the trivial representation of Q in
the spectrum of α is 1.
We will often refer to B as a quantum space. It can be easily shown that a
quantum space is homogeneous if and only if the corresponding co-action is
ergodic (i.e. α(x) = x ⊗ I implies x is a scalar multiple of the identity of B.
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4
Rieffel-Wang Deformation of compact quantum
groups
Let θ = ((θkl )) be a skew symmetric matrix of order n. We denote by C(Tnθ )
the universal C ∗ -algebra generated by n unitaries (U1 , U2 , ...Un ) satisfying
Uk Ul = e2πθkl Ul Uk , for k 6= l. Caution: this is not a commutative C ∗ algebra of continuous functions of some space in general, and C(Tnθ ) is just
a convenient notation to suggest that it comes from Tn by a deformation
depending on the metrix-parameter θ. If θkl = θ0 for k < l, where θ0 ∈ R,
we will denote the corresponding universal C ∗ -algebra by C(Tnθ0 ) and W will
denote the ∗-subalgebra generated by unitaries U1 , U2 , ...Un .
Let A be a unital C ∗ -algebra on which there is a strongly continuous ∗automorphic action σ of Tn . Denote by τ the natural action of Tn on C(Tnθ )
given on the generators Ui0 s by τ (z)Ui = zi Ui , where z = (z1 , z2 , ...zn ) ∈ Tn .
Let τ −1 denote the inverse action s → τ−s .
Definition 4.1 The fixed point algebra of A ⊗ C(Tnθ ), under the action (σ ×
−1
τ −1 ), i.e. (A ⊗ C(Tnθ ))σ×τ , is called the Rieffel deformation of A under
the action σ of Tn , and is denoted by Aθ .
Rieffel’s original approach to define the above was somewhat different. He
defined a ‘twisted’ multiplication resembling convolution on the subalgebra
A∞ ⊆ A of smooth elements for the given action of Tn . By completing A∞
w.r.t. a suitable norm one gets the deformed C ∗ algebra.
There is a natural isomorphism between (Aθ )−θ and A, given by the
−1
identification of A with the subalgebra of (A ⊗ C(Tnθ ) ⊗ C(Tnθ ))(σ⊗id)×τ
generated by elements of the form ap ⊗ U p ⊗ (U 0 )p , where p = (p1 , p2 , ...pn ) ∈
Zn , U p := U1p1 U2p2 ...Unpn , (U 0 )p := (U10 )p1 (U20 )p2 ....(Un0 )pn , U10 , U20 , ...Un0 being
the generators of C(Tn−θ ) and ap belongs to the spectral subspace of the
action σ corresponding to the character p.
Let Q be a CQG with coproduct ∆ and assume that there exists a
surjective CQG morphism π : Q → C(Tn ) which identifies C(Tn ) as a
quantum subgroup of Q. For s ∈ Tn , let Ω(s) denote the state defined by
Ω(s) := evs ◦π, where evs denotes evaluation at s. Define an action of T2n on
Q by (s, u) → χ(s,u) , where χ(s,u) := (Ω(s) ⊗ id) ◦ ∆ ◦ (id ⊗ Ω(−u)) ◦ ∆. It has
been shown by Wang that the Rieffel deformation Qθ,−
e θe of Q with respect
0 θ
to θe :=
can be given a unique CQG structure such that the
−θ 0
4
the Hopf∗ algebra of Qθ,−θ is isomorphic as a coalgebra with the cannonical
Hopf-∗ algebra of Q.
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