SIMPLE COMPACT QUANTUM GROUPS I

SHUZHOU WANG

Abstract. The notion of simple compact quantum groups is introduced and some basic properties are established. As non-trivial (i.e. noncommutative and noncocommutative) examples, the following compact quantum groups are shown to be simple: The universal quantum groups B_{u}(Q) for Q ∈ GL(n, ℂ) satisfying QQ̄ = ±I_{n}, n ≥ 2; The quantum automorphism groups A_{aut}(B, τ) of finite dimensional C^{∗}-algebras B endowed with the canonical trace τ when dim(B) ≥ 4, including the quantum permutation groups A_{aut}(X_{n}) on n points (n ≥ 4); The standard deformations K_{q} of simple compact Lie groups K and their twists K_{q}^{u}, as well as Rieffel’s deformation K_{J}.

1. Introduction

The theory of quantum groups saw spectacular breakthroughs in the 1980’s when on the one hand Drinfeld and Jimbo discovered the quantized universal enveloping algebras of semisimple Lie algebras [14, 18] based on the work of the Faddeev school on the quantum inverse scattering method, on the other hand Woronowicz independently discovered quantum deformations of compact Lie groups and formulated the axioms for compact quantum groups [51, 52, 53]. Moreover, further work of Rosso, Soibelman and Vaksman, Levendorskii [29, 31, 34, 33, 21] showed that the Drinfeld-Jimbo quantum groups admit “compact real forms” K_{q} and twists K_{q}^{u} that fit into the framework of Woronowicz, most notable of which is the work of Soibelman [33], in which a unified theory of deformations of simple compact Lie groups is developed based on his earlier joint work with Vaksman [34].

Starting in his Ph.D. thesis [39], the author of the present article took a different direction from the above by viewing quantum groups as intrinsic objects and found in a series of papers (including [38] in collaboration with Van Daele) several classes of compact quantum groups that can not be obtained as deformations of Lie groups. The most important of these are the universal compact quantum groups of Kac type A_{u}(n) and their self-conjugate counterpart A_{o}(n) [40], the more general universal compact quantum groups A_{u}(Q) and their self-conjugate counterpart B_{u}(Q) [38, 44], where Q ∈ GL(n, ℂ), and the quantum automorphism groups A_{aut}(B, tr) of finite dimensional C^{∗}-algebras B endowed with a tracial functional tr, including the quantum permutation groups A_{aut}(X_{n}) on the space X_{n} of n points. Further studies of these quantum groups reveal remarkable properties. We mention a few.

(1) According to deep work of Banica [2, 3, 4], the representation rings (also called the fusion rings) of the quantum groups B_{u}(Q) and A_{aut}(B, τ) are isomorphic to those of appropriate compact Lie groups when QQ̄ is a scalar, dim(B) ≥ 4 (τ being the canonical trace on B), and the representation ring of A_{u}(Q) is almost a free product of two copies of ℤ; (2) The compact quantum groups A_{u}(Q) admit ergodic actions on both finite and infinite injective von Neumann factors and the quantum groups A_{o}(n) admit minimal (i.e. prime) actions on the Murray-von Neumann factor R [46]; (3) The special A_{u}(Q)’s for positive Q and B_{u}(Q)’s for Q satisfying the property QQ̄ = ±1 have tractable isomorphic classifications and the general A_{u}(Q)’s and B_{u}(Q)’s for arbitrary Q can be decomposed explicitly as free products of the former special ones [48]; (4) The quantum permutation groups A_{aut}(X_{n}) admit interesting quantum subgroups that appear in connection with other areas of mathematics, such as the quantum automorphism groups of finite graphs and the free wreath products discovered by Bichon [7, 8], as well as certain quantum symmetry groups in the theory of subfactors studied by Banica [6].

1991 Mathematics Subject Classification. Primary 46L65, 46L87, 46L55, 46L60, 46L89, 58B32; Secondary 16W30, 17B37, 20G42, 81R50, 81R60.

Key words and phrases. Simple Quantum Groups, Noncommutative Geometry, Quantizations, Deformations, Woronowicz C^{∗}-algebras, Hopf Algebras.

Research supported in part by the National Science Foundation grant DMS-0096136.

The purpose of this article is to initiate a study of simple compact quantum groups and show that the compact quantum groups mentioned in the last two paragraphs are simple in generic cases. An outline of its contents is as follows. From §2 to §4, we develop several basic notions related to simple compact quantum groups. In §2, we prove several equivalent conditions for a quantum subgroup of a compact quantum group to be normal, a notion introduced in [39, 40]. For more on the notion of normal quantum subgroups, we refer the reader to [49]. In §3, we develop some basic properties of normal quantum subgroups. In §4 the notion of simple compact quantum groups is introduced. A compact matrix quantum is called simple if it is connected and has no non-trivial connected normal quantum subgroups and no non-trivial representations of dimension one. We also introduce in this section quantum groups with property F and those with property F D, as well as almost classical compact quantum groups. In connection with noncommutative geometry, compact quantum groups with property F and almost classical compact quantum groups are two classes of quantum groups closest to compact Lie groups, though they need not be quantum deformations of the latter. All of the simple compact quantum groups studied in this paper belong to both classes. After preparatory work in the first few sections, the main examples of this paper are described in the last two sections, §5 and §6. Recall [4] that the canonical trace τ on a finite dimensional C^{∗}-algebra B is the restriction of the unique tracial state on the algebra L(B) of operators on B. In §5, we prove the main results on B_{u}(Q) and A_{aut}(B, τ) (cf. Theorem 5.1 and Theorem 5.5):

Theorem 1.1. Let Q ∈ GL(n, ℂ) be such that QQ̄ = ±I_{n} and n ≥ 2. Then B_{u}(Q) is a simple compact quantum group with property F.

Theorem 1.2. Let B be a finite dimensional C^{∗}-algebra with dim(B) ≥ 4 and τ its canonical trace. Then A_{aut}(B, τ) is an absolutely simple compact quantum group with property F.

Besides the correspondence between Hopf ∗-ideals and Woronowicz Hopf C^{∗}-ideals which we develop in §5, the proofs of these two results rely heavily on the fundamental work of Banica on the structure of fusion rings of these quantum groups [2, 4]. The last section §6 is devoted to the standard deformations K_{q} of simple compact Lie groups, their twists K_{q}^{u} [33, 21, 22], and Rieffel’s quantum groups K_{J} [28], where q ∈ ℝ\{0}, u ∈ ∧^{2}(i𝔱) and J is an appropriate skew-symmetric transformation on the direct sum 𝔱 ⊕ 𝔱 of Cartan subalgebra 𝔱 of the Lie algebra of K (cf. Theorem 6.1, Theorem 6.6 and Theorem 6.4):

Theorem 1.3. Let K be a connected and simply connected simple compact Lie group. Then both K_{q} and its twists K_{q}^{u} are simple compact quantum groups with property F.

Theorem 1.4. Let K be a simple compact Lie group with a toral subgroup T of rank at least two. Then K_{J} is a simple compact quantum group with property F.

The proofs of Theorem 1.3 and Theorem 1.4 make use of the work of Lusztig and Rosso [23, 30] on representations of quantized universal enveloping algebras, the work of Soibelman and Levendorskii [33, 21, 22] on quantum function algebras of K_{q} and K_{q}^{u}, and the work of Rieffel [28] and the author [43] on strict deformations of Lie groups and quantum groups.

Classification of simple compact quantum groups and their irreducible representations up to isomorphism are two of the main goals in the study of compact quantum groups. Namely, one would like to develop a theory of simple compact quantum groups that parallels the Killing-Cartan theory and the Cartan-Weyl theory for simple compact Lie groups. To accomplish the first goal, one must first construct all simple compact quantum groups. Though we have given several infinite classes of examples of these in this article, it should be pointed out that the construction of simple compact quantum groups is far from being complete. In fact it is fair to say that we are only at the beginning stage for this task at the moment. One indication of this is that all the simple compact quantum groups known so far are almost classical in the sense that their representation rings are isomorphic to those of ordinary compact groups and are in particular commutative. The first candidates of simple compact quantum groups that are not almost classical would be A_{u}(Q), where Q ∈ GL(n, ℂ) are positive, n ≥ 2. The representation ring of A_{u}(Q) is “very” noncommutative, being close to the free product of two copies of the ring of integers, according to Banica [3]. To construct the other (“the remaining”) simple compact quantum groups, we believe the most profitable approach is to study quantum automorphism groups of appropriate quantum spaces, such as those in our papers [45, 46, 47] and the papers of Banica and Bichon [6, 7, 8]. In retrospect, both simple Lie groups and finite simple groups are automorphism groups, so it is natural to take a similar approach for the theory of simple quantum groups.

Convention and Notation. We assume that all Woronowicz C^{∗}-algebras (also called Woronowicz Hopf C^{∗}-algebras) considered in this paper to be full unless otherwise explicitly stated, since morphisms between quantum groups are meaningful only for full Woronowicz C^{∗}-algebras cf. [40, 42]. For a compact quantum group G, A_{G} denotes the underlying Woronowicz C^{∗}-algebra and A_{G} the associated dense Hopf ∗-algebra of quantum representative functions on G. Sometimes we also call A_{G} a compact quantum group, referring to G. See [40, 52] for more on other unexplained definitions and notations used in this paper.

2. The Notion of Normal Quantum Subgroups

Before making the notion of simple quantum groups precise, we study further the notion of normal quantum subgroups (of compact quantum groups) introduced in [39, 40]. Let (N, π) be a quantum subgroup of a compact quantum group G with surjections π : A_{G} ⟶ A_{N} and π̂ : A_{G} ⟶ A_{N}. The quantum group (N, π) should be more precisely called a closed quantum subgroup, but we will omit the word closed in this paper, since we do not consider non-closed quantum subgroups. Define

A_{G/N} = {a ∈ A_{G}|(id ⊗ π)Δ(a) = a ⊗ 1_{N}},
A_{N\G} = {a ∈ A_{G}|(π ⊗ id)Δ(a) = 1_{N} ⊗ a},

where Δ is the coproduct on A_{G}, 1_{N} is the unit of the algebra A_{N}. We omit the subscript N in 1_{N} when no confusion arises. Similarly, we define

A_{G/N} = A_{G} ∩ A_{G/N}, and A_{N\G} = A_{G} ∩ A_{N\G}.

Note that G/N, N\G shall be denoted more precisely by G/(N, π), (N, π)\G respectively, if there is a possible confusion. Let h_{N} be the Haar measure on N. Let

E_{G/N} = (id ⊗ h_{N}π)Δ, E_{N\G} = (h_{N}π ⊗ id)Δ.

Then E_{G/N} and E_{N\G} are projections of norm one (completely positive and completely bounded conditional expectations) from A_{G} onto A_{N\G} and A_{G/N} respectively (cf. [25] as well as Proposition 2.3 and Section 6 of [46]), and

A_{G/N} = E_{G/N}(A_{G}), and A_{N\G} = E_{N\G}(A_{G}).

From this, we see that the *-subalgebras A_{N\G} and A_{G/N} are dense in A_{G/N} and A_{N\G} respectively.

Assume N is a closed subgroup of an ordinary compact group G. Let π be the restriction morphism from A_{G} := C(G) to A_{N} := C(N). Let C(G/N) and C(N\G) be continuous functions on G/N and N\G respectively. Then one can verify that

C(G/N) = A_{G/N} = E_{G/N}(A_{G}),
C(N\G) = A_{N\G} = E_{N\G}(A_{G}).

Therefore we will use the symbols C(G/N) and A_{G/N} (resp. C(N\G) and A_{N\G}; C(G) and A_{G}) interchangeably for all quantum groups. Proposition 2.1. Let N be a quantum subgroup of a compact quantum group G. Then the following conditions are equivalent:

(1) A_{N\G} is a Woronowicz C^{∗}-subalgebra of A_{G}.

(2) A_{G/N} is a Woronowicz C^{∗}-subalgebra of A_{G}.

(3) A_{G/N} = A_{N\G}.

(4) For every irreducible representation u^{λ} of G, either h_{N}π(u^{λ}) = I_{d_{λ}} or h_{N}π(u^{λ}) = 0, where h_{N} is the Haar measure on N, d_{λ} is the dimension of u^{λ} and I_{d_{λ}} is the d_{λ} × d_{λ} identity matrix.

Proof. We only need to show that (1)⇔(4)⇔(3). The proof of the implications (2)⇔(4)⇔(3) is similar.

(3)⇒(4): In general one has

Δ(A_{N\G}) ⊆ A_{N\G} ⊗ A_{G}, Δ(A_{G/N}) ⊆ A_{G} ⊗ A_{G/N}.

Letting B = A_{N\G} = A_{G/N} then one has

Δ(B) ⊆ B ⊗ B.

For λ ∈ Ĝ, let n_{λ} be the multiplicity of the trivial representation of N in the representation π(u^{λ}). We claim that either n_{λ} = d_{λ} or n_{λ} = 0.

Assume on the contrary that there is a λ ∈ Ĝ such that 1 < n_{λ} < d_{λ}. Note that in general

E_{N\G}(u_{ij}^{λ}) = (h_{N}π ⊗ id)Δ(u_{ij}^{λ}) = ∑_{k} h_{N}π(u_{ik}^{λ})u_{kj}^{λ},

E_{G/N}(u_{ij}^{λ}) = (id ⊗ h_{N}π)Δ(u_{ij}^{λ}) = ∑_{k} h_{N}π(u_{kj}^{λ})u_{ik}^{λ}.

Using unitary equivalence if necessary we choose u_{ij}^{λ} in such a way that the n_{λ} trivial representations of N appear on the upper left diagonal corner of π(u^{λ}). Then

E_{N\G}(u_{ij}^{λ}) = \begin{cases}u_{ij}^{λ} & if 1 ≤ i ≤ n_{λ}, 1 ≤ j ≤ d_{λ}, \\ 0 & if n_{λ} < i ≤ d_{λ}, 1 ≤ j ≤ d_{λ}, \end{cases}

E_{G/N}(u_{ij}^{λ}) = \begin{cases}u_{ij}^{λ} & if 1 ≤ i ≤ d_{λ}, 1 ≤ j ≤ n_{λ}, \\ 0 & if 1 ≤ i ≤ d_{λ}, n_{λ} < j ≤ d_{λ}. \end{cases}

Since A_{N\G} = A_{G/N} = B and both E_{N\G} and E_{G/N} are projections from A_{G} onto B, we have

E_{N\G} = E_{G/N}.

Then for n_{λ} < j ≤ d_{λ},

0 ≠ u_{ij}^{λ} = E_{N\G}(u_{ij}^{λ}) = E_{G/N}(u_{ij}^{λ}) = 0,

This is a contradiction.

(4)⇒(3): Let S(N) (or S(N, π)) be the subset of Ĝ consisting of those λ’s such that h_{N}π(u^{λ}) is I_{d_{λ}}. Then a straightforward calculation using the fact that E_{N\G} and E_{G/N} are projections of A_{G} onto A_{N\G} and A_{G/N} respectively, one gets

A_{N\G} = A_{G/N} = ⨁{ℂu_{ij}^{λ} | λ ∈ S(N), i, j = 1, ⋯, d_{λ}}.

(4)⇒(1): Let S(N) be defined as in the proof of (4)⇒(3). It is clear that A_{N\G} is a Woronowicz C^{∗}-subalgebra of A_{G} and that

{u_{ij}^{λ} | λ ∈ S(N), i, j = 1, ⋯, d_{λ}}.

is a Peter-Weyl basis of the dense ∗-subalgebra A_{N\G} of A_{N\G}.

(1)⇒(4): Let G_{1} = N\G. Then by Woronowicz’s Peter-Weyl theorem for compact quantum groups, every irreducible representation u^{λ} of G is either an irreducible representation of G_{1} or none of the coefficients u_{ij}^{λ} is in A_{G_{1}}. That is

E_{N\G}(u_{ij}^{λ}) = \begin{cases}u_{ij}^{λ} & if λ ∈ Ĝ_{1}, \\ 0 & if λ ∈ Ĝ\Ĝ_{1}. \end{cases}

By the definition of E_{N\G} and linear independence of the u_{ij}^{λ}’s, this implies that

h_{N}π(u_{ik}^{λ}) = δ_{ik}, λ ∈ Ĝ_{1}, i, k = 1, ⋯, d_{λ},
h_{N}π(u_{ik}^{λ}) = 0, λ ∈ Ĝ\Ĝ_{1}.

This completes the proof of the proposition. □ Definition 2.2. A quantum subgroup N of a compact quantum group G is said to be normal if it satisfies the equivalent conditions of Proposition 2.1. Remarks. (a) Condition (4) of Proposition 2.1 plays an important role in this paper. It is a reformulation of the following condition for a normal quantum subgroup N that appears near the end of Sect. 2 of [40]: For every irreducible representation v of G, the multiplicity of the trivial representation of N in the representation π(v) is either zero or the dimension of v. From the proof of the proposition we see that the counit of A_{G/N} is equal to the restriction morphism π|_{A_{G/N}}.

(b) Note also that on p679 of [40] the following statement is found: “In general, a right quotient quantum group is different from the corresponding left quotient quantum group.” Though in the purely algebraic setting of Hopf algebras, one needs to distinguish between left and right normal quantum subgroups, as indicated in 1.5 of Parshall and Wang [24] (see also [1, 32, 36]), however, in view of Proposition 2.1 above, this cannot happen for normal quantum subgroups of compact quantum groups. We will see in Lemma 5.3 later that when applied to compact quantum groups the notion of normality defined in [24] is equivalent to our notion of normality. For compact quantum groups, one can also verify the conditions in 1.5.1.(1) of [24], so the notions of left and right normal quantum subgroups in their sense coincide. We note further that the antipode for compact quantum groups is not continuous and is not defined on the C^{∗}-algebra level in general. As a result one cannot naively carry those definitions involving the antipode in the Hopf algebras context, such as the notion of normality in [24], over to compact quantum groups. (See Lemma 5.2 and Lemma 5.3 later.)

(c) The notion of a normal quantum subgroup depends on the morphism π, which gives the “position” of the quantum group N in G. If (N, π_{1}) is another quantum subgroup of G with surjection π_{1} : A_{G} ⟶ A_{N}, (N, π_{1}) may not be normal even if (N, π) is. This phenomenon already occurs in the group situation. For example a finite group can contain two isomorphic subgroups with one normal but the other not.

Examples. We show in (1) and (2) below that the identity group and the full quantum group G are both normal quantum subgroups of G under natural embeddings. These will be called the trivial normal quantum subgroups.

(1) Let N = {e} be the one element identity group. Let π = ϵ = counit of A_{G} be the morphism from A_{G} to A_{N}. Then by the counital property, one has

A_{G/N} = {a ∈ A_{G}|(id ⊗ ϵ)Δ(a) = a ⊗ 1} = A_{G}.

That is ({e}, ϵ) is normal and G/({e}, ϵ) = G.

(2) Now let N = G and let π : A_{G} → A_{N} be any isomorphism of Woronowicz C^{∗}-algebras [40]. Let h be the Haar measure on G and a ∈ A_{G/N}. Since π is an isomorphism and (id ⊗ π)Δ(a) = a ⊗ 1, one has Δ(a) = (id ⊗ π)^{−1}(a ⊗ 1) = a ⊗ 1. From the invariance of h and one has

h(a)1 = (1 ⊗ h)Δ(a) = ah(1) = a.

Hence

A_{G/N} = {a ∈ A_{G}|(id ⊗ π)Δ(a) = a ⊗ 1} = ℂ1.

That is (G, π) is normal and G/(G, π) ≅ {e}.

(3) We note that besides the embeddings in (2) it is possible to construct examples of compact quantum groups G with non-normal proper embeddings of G into G. In fact this can happen for compact groups already. □

The following is a justification of the above notion of normal quantum subgroups. Proposition 2.3. Let A = C(G) with G a compact group. Let N be a closed subgroup of G. Let π be the restriction map from A to A_{N} = C(N). Then (N, π) is normal in the sense above if and only if N is a normal subgroup of G in the usual sense.

Proof. Under the Gelfand-Naimark correspondence which associates to every commutative C^{∗}-algebra its spectrum, quotients of G by (ordinary) closed normal subgroups N correspond to Woronowicz C^{∗}-subalgebras of C(G), i.e.,

G/N corresponds to C(G/N),

see 2.6 and 2.12 of [40]. Since A_{G/N} = C(G/N) for any closed subgroup N, the proposition follows from Proposition 2.1 above. □

The following result gives a complete description of quantum normal subgroups of the compact quantum group dual of a discrete group Γ, whose proof is straightforward using e.g. [52] and Proposition 2.1. Proposition 2.4. Let A_{G} = C^{∗}(Γ). Let N be a quantum subgroup of G with surjection π : A_{G} → A_{N}. Then N is normal, L := π(Γ) is a discrete group and A_{N} = C^{∗}(L). Moreover, A_{G/N} = C^{∗}(K), where K = ker(π : Γ → L).

To distinguish two different quantum subgroups, we include the following result, which should be known to experts in the theory of C^{∗}-algebras. Proposition 2.5. Let π_{k} : A → A_{k} be surjections of unital C^{∗}-algebras with kernels I_{k} (k = 1, 2). Let P_{k} be the pure state space of A_{k}. Then the following conditions are equivalent:

(1) {ϕ_{1} ∘ π_{1}|ϕ_{1} ∈ P_{1}} = {ϕ_{2} ∘ π_{2}|ϕ_{2} ∈ P_{2}} as subsets of pure states of A.

(2) I_{1} = I_{2}.

(3) There is an isomorphism α : A_{1} → A_{2} such that π_{2} = α ∘ π_{1}.

Proof. (1) ⇒ (2): If I_{1} ≠ I_{2}, say, there is a x ∈ I_{1}\I_{2}. Then there is a pure state ϕ of A/I_{2} such that ϕ(π_{2}(x)) ≠ 0, where we identify A_{2} with A/I_{2}. But ϕπ_{2} is a pure state of A/I_{1} ≅ A_{1} according to assumption (1). Hence we must have ϕπ_{2}(x) = 0. This is a contradiction.

(2) ⇒ (3): Let I = I_{1} = I_{2}. Let π be the quotient map A → A/I. Let π̃_{k} be the homomorphism from A/I to A_{k} such that π_{k} = π̃_{k}π (k = 1, 2). Then π̃_{k} are isomorphisms. Put α = π̃_{2} ∘ π̃_{1}^{−1}. Then π_{2} = α ∘ π_{1}.

(3) ⇒ (1): This follows from P_{1} = P_{2} ∘ α. □

The following proposition is an easy consequence of Proposition 2.1. Proposition 2.6. Let (N_{1}, π_{1}) be a normal quantum subgroup of G. Let

α : A_{N_{1}} ⟶ A_{N_{2}}

be an isomorphism of quantum groups. Then (N_{2}, απ_{1}) is normal.

In view of the above discussions, it is reasonable to have the following definition (cf. also remarks after Proposition 2.3). Definition 2.7. Two quantum subgroups (π_{1}, H_{1}) and (π_{2}, H_{2}) of a quantum group G are called the same if π_{1}, π_{2} satisfy the equivalent conditions of Proposition 2.5. When this happens, we denote (π_{1}, H_{1}) = (π_{2}, H_{2}).

Geometrically speaking, two “embedded” quantum subgroups (π_{1}, H_{1}) and (π_{2}, H_{2}) of a quantum group G are called the same if their “images” in G are the same.

3. Properties of Normal Quantum Subgroups

Parallel to the three fundamental isomorphism theorems in the theory of groups, one would expect that their analogs are valid in the theory of quantum groups. It is difficult to formulate appropriate quantum analogs of the first two isomorphism theorems, cf. [24, 32, 36, 1]. The following is an analog of the third fundamental isomorphism theorem. Proposition 3.1. Let (N, ρ) be a normal quantum subgroup of G. Let (H, π) be a quantum subgroup of G that contains (N, ρ), i.e., there is a morphism π_{1} such that (N, π_{1}) is a quantum subgroup of H with ρ = π_{1}π. Then (N, π_{1}) is normal in H. If furthermore H is normal in G, then (H/N, π′) is normal in G/N and

A_{(G/N)/(H/N)} = A_{G/H},

where π′ = π|_{A_{G/N}}, the restriction of π to A_{G/N}.

Proof. According to the remarks near the beginning of §2, we only need to work with the dense ∗-subalgebras.

Let z ∈ A_{H} be such that π_{1}(z) = 0. Assume z = π(x) for some x ∈ A_{G}. Then ρ(x) = 0. Since ker(ρ̂) is normal in the sense of [32] by remark (b) following Lemma 5.3 later, we have

∑ ρ(x_{(2)}) ⊗ S(x_{(1)})x_{(3)} = 0, ∑ ρ(x_{(2)}) ⊗ x_{(1)}S(x_{(3)}) = 0,

where we use Sweedler’s summation convention [35]. Hence

∑ π_{1}π(x_{(2)}) ⊗ π(S(x_{(1)})x_{(3)}) = 0, ∑ π_{1}π(x_{(2)}) ⊗ π(x_{(1)}S(x_{(3)})) = 0.

That is

∑ π_{1}(z_{(2)}) ⊗ S(z_{(1)})z_{(3)} = 0, ∑ π_{1}(z_{(2)}) ⊗ z_{(1)}S(z_{(3)}) = 0.

This means that (π_{1}, N) is normal in H.

Let a ∈ A_{G/N}. Then it is clear that π(a) ∈ A_{H/N} so π(A_{G/N}) is contained in A_{H/N}. Conversely, let b ∈ A_{H/N}. Assume b = π(a) for some a ∈ A_{G}. Put ā = (1 ⊗ h_{N}ρ)Δ(a). Then one verifies that ā ∈ A_{G/N} (cf. Proposition 2.3 and Sect. 6 of [46]) and π(ā) = b. That is π(A_{G/N}) = A_{H/N}.

Now assume (H, π) is normal. For ease of notation, Let G′ = G/N and H′ = H/N. The above shows that (H′, π′) is a quantum subgroup of G′. If a ∈ A_{G′/H′}, that is, (1 ⊗ π′)Δ(a) = a ⊗ 1_{H′}, then it is clear that a ∈ A_{G/H} since π′ = π|_{A_{G/N}} and 1_{H′} = 1_{H}. Conversely, if a ∈ A_{G/H}, that is (1 ⊗ π)Δ(a) = a ⊗ 1_{H}, then (1 ⊗ π_{1}π)Δ(a) = a ⊗ 1_{N}. That is a ∈ A_{G′} = A_{G/N}. But then we have

(1 ⊗ π′)Δ(a) = (1 ⊗ π)Δ(a) = a ⊗ 1_{H} = a ⊗ 1_{H′}.

Hence a ∈ A_{G′/H′} and A_{G′/H′} = A_{G/H}.

The result is completely proved. □

Just as in the case of ordinary groups, we have Proposition 3.2. Let G_{1}, G_{2} be compact quantum groups. Let π_{1} be the natural embedding of G_{1} into G_{1} × G_{2} defined by the surjection

π_{1} : A_{G_{1}} ⊗ A_{G_{2}} → A_{G_{1}},
π_{1} = id_{1} ⊗ ϵ_{2}, ϵ_{2} being the counit of A_{G_{2}}.

Then (G_{1}, π_{1}) is a normal quantum subgroup of G = G_{1} × G_{2} and G/G_{1} ≅ G_{2}.

Proof. According to [41], every irreducible representation of G_{1} × G_{2} is of the form

u ⊗_{ex} v = ∑_{ijkl} e_{ij}^{u} ⊗ e_{kl}^{v} ⊗ ⊗u_{ij} ⊗ v_{kl},

where u and v are irreducible representations of G_{1} and G_{2} respectively, e_{ij}^{u} and e_{kl}^{v} are systems of matrix units acting on the Hilbert spaces of u and v respectively. Let h_{1} be the Haar state of G_{1}. Then

h_{1}π_{1}(u ⊗_{ex} v) = ∑_{ijkl} e_{ij}^{u} ⊗ e_{kl}^{v}h_{1}(u_{ij})δ_{kl} = ∑_{ij} e_{ij}^{u}h_{1}(u_{ij}) ⊗ I_{v},

which is either I_{u} ⊗ I_{v} or 0 according as u is trivial or not, where I_{u}, I_{v} are the identity matrices acting on the Hilbert spaces of u, v respectively. By Proposition 2.1.(4), G_{1} is normal.

Similarly, we have

E_{G/G_{1}}(u_{ij} ⊗ v_{kl}) = \begin{cases}0 & if u is non-trivial, \\ 1 ⊗ v_{kl} & if u is trivial. \end{cases}

Hence A_{G/G_{1}} = 1 ⊗ A_{G_{2}} ≅ A_{G_{2}}. □

For free products [40], the situation is quite different: Proposition 3.3. Let G_{1}, G_{2} be compact quantum groups. Let G = G_{1}ˆ∗G_{2} be the free product compact quantum group underlying A_{G_{1}} ∗ A_{G_{2}}. Let π_{1} be the natural embedding of G_{1} into G defined by the surjection

π_{1} : A_{G_{1}} ∗ A_{G_{2}} → A_{G_{1}},
π_{1} = id_{1} ∗ ϵ_{2}.

If G_{1} has at least one irreducible representation of dimension greater than one, then (G_{1}, π_{1}) is not a normal quantum subgroup of G_{1}ˆ∗G_{2}. Otherwise, (G_{1}, π_{1}) is normal in G_{1}ˆ∗G_{2}.

The hat in the symbol ˆ∗ above signifies the “Fourier transform” of ∗.

Proof. Let

u = ∑_{ij} e_{ij}^{u} ⊗ u_{ij}, v = ∑_{kl} e_{kl}^{v} ⊗ v_{kl}

be irreducible representations of G_{1}, G_{2} respectively. Assume that the dimension of u is greater than one. Let ū = ∑_{ij} e_{ij}^{u} ⊗ u_{ij}^{∗} denote the conjugate representation of u. Then by [40], the interior tensor product representation

u ⊗_{in} v ⊗_{in} ū = ∑_{ijklrs} e_{ij}^{u} ⊗ e_{kl}^{v} ⊗ e_{rs}^{u} ⊗ u_{ij}v_{kl}u_{rs}^{∗}

is irreducible. Let h_{1} be the Haar state of G_{1}. Then

h_{1}π_{1}(u ⊗_{in} v ⊗_{in} ū) = ∑_{ijrs} e_{ij}^{u} ⊗ I_{v} ⊗ e_{rs}^{u}h_{1}(u_{ij}u_{rs}^{∗}),

where I_{v} is the identity matrix acting on the Hilbert space of v. Since u is of dimension greater than one, u ⊗ ū properly contains the trivial representation of G_{1} (with multiplicity one). Hence

h_{1}(u ⊗_{in} ū) = ∑_{ijrs} e_{ij}^{u} ⊗ e_{rs}^{u}h_{1}(u_{ij}u_{rs}^{∗})

is neither the identity matrix, nor the zero matrix on H_{u} ⊗ Hū. (See also Theorem 5.7 of Woronowicz [52].) Therefore

h_{1}π_{1}(u ⊗_{in} v ⊗_{in} ū)

is neither the identity matrix, nor the zero matrix on H_{u} ⊗ H_{v} ⊗ Hū. By Proposition 2.1.(4), G_{1} is not normal.

If G_{1} has no non-trivial irreducible representations of dimension greater than one, then by [52] G_{1} is the compact quantum group dual of a discrete group Γ, i.e., A_{G_{1}} = C^{∗}(Γ). By [40], every irreducible representation of G is of the form

w^{λ_{1}} ⊗ w^{λ_{2}} ⊗ ⋯ ⊗ w^{λ_{n}},

where w^{λ_{i}} is a non-trivial representation belonging to either the set Γ or the set Ĝ_{2}, w^{λ_{i}} and w^{λ_{i+1}} being in different sets. It is clear that π_{1}(w^{λ_{1}} ⊗ w^{λ_{2}} ⊗ ⋯ ⊗ w^{λ_{n}}) is a constant diagonal matrix, with diagonal entry (call it [w^{λ_{1}} ⊗w^{λ_{2}} ⊗⋯⊗w^{λ_{n}}]) equal to the product of those w^{λ_{i}}’s that belong to Γ. Since h_{G_{1}}(γ) = 0 if γ is not the neutral element of Γ, one sees that Proposition 2.1.(4) is fulfilled. Hence (G_{1}, π_{1}) is normal in G = G_{1}ˆ∗G_{2} and A_{G/G_{1}} is equal to the closure of the linear span of entries of the matrix w^{λ_{1}} ⊗ w^{λ_{2}} ⊗ ⋯ ⊗ w^{λ_{n}} such that [w^{λ_{1}} ⊗ w^{λ_{2}} ⊗ ⋯ ⊗ w^{λ_{n}}] is the neutral element of Γ. □

Similar to the second situation in the proposition above, we have the following Proposition 3.4. Consider a crossed product A ⋊_{α} Γ of a compact quantum group A by a discrete group Γ [41]. Then C^{∗}(Γ) is a quantum normal subgroup of A ⋊_{α} Γ with quotient A via the morphism

π : A ⋊_{α} Γ ⟶ C^{∗}(Γ), π(aγ) = ϵ(a)γ,

where ϵ is the counit of A and a ∈ A, γ ∈ Γ.

More generally, one has Proposition 3.5. Let (A, Γ, α) be a Woronowicz C^{∗}-dynamical system. Let I ⊲ A be an α-invariant Woronowicz C^{∗}-ideal so that A/I is a normal quantum subgroup of A with quotient quantum group B. Let K be the kernel of the action α. Let α̃ be the induced action of Γ/K on A/I. Then A/I ⋊α̃ Γ/K is a normal quantum subgroup of A ⋊_{α} Γ with quotient B ⋊_{α} K.

Proof. Consider the morphisms

π : A ⟶ A/I ⋊α̃ Γ/K,
U : Γ ⟶ A/I ⋊α̃ Γ/K,

defined by π(a) = ã, U_{γ} = γ̃. Here ã is the image in A/I of an element a ∈ A viewed as an element of A/I ⋊α̃ Γ/K via inclusion; γ̃ is the image in Γ/K of an element γ ∈ Γ viewed as an element of A/I ⋊α̃ Γ/K via inclusion. One can verify that (π, U) is a covariant representation, i.e.,

U_{γ}π(a)U_{γ}^{−1} = π(α_{γ}(a)).

Hence there is a surjection

π × U : A ⋊_{α} Γ ⟶ A/I ⋊α̃ Γ/K,

extending π and U. This morphism preserves the coproducts because its restrictions to A and C^{∗}(Γ) do. This shows that A/I ⋊α̃ Γ/K is a quantum subgroup of A ⋊_{α} Γ (true under only the assumption that A/I is a quantum subgroup of A).

Now assume that A/I is a normal quantum subgroup of A with quotient B. Let θ denote the surjection π × U. Let h = h_{A/I} ⋊α̃ h_{Γ/K} be the Haar state on A/I ⋊α̃ Γ/K (cf. [41]). Then every irreducible representation of the quantum group A ⋊_{α} Γ is of the form (u_{ij}^{β}γ), where (u_{ij}^{β}) is an irreducible representation of dimension d_{β} of the quantum group A and γ ∈ Γ (cf. [41] as well). Let S be as in proof of (4)⇒(3). Then

(hθ(u_{ij}^{β}γ)) = (h(ũ_{ij}^{β})h(γ̃)) = \begin{cases}I_{d_{β}} & if β ∈ S, γ ∈ K, \\ 0 & otherwise. \end{cases}

Hence by Proposition 2.1 and its proof, we conclude that A/I ⋊α̃ Γ/K is a normal quantum subgroup of A ⋊_{α} Γ with quotient B ⋊_{α} K. □

We note that A is not a normal quantum subgroup of A⋊_{α} Γ, unlike the semi-direct product of groups.

4. Simple Compact Quantum Groups

To avoid undue difficulties in the theory of simple compact quantum groups, we assume connectivity as a part of their postulates. The latter concept is given by Definition 4.1. We call a compact quantum group G_{A} connected if for each nontrivial irreducible representation u^{λ} ∈ Ĝ_{A}, the C^{∗}-algebra C^{∗}(u_{ij}^{λ}) generated by the coefficients of u^{λ} is of infinite dimension.

In virtue of (28.21) of [17], we have Proposition 4.2. Let G_{A} be an ordinary compact group (i.e. A_{G} is commutative). Then G_{A} is connected as a topological space if and only if it is connected in the sense above. Definition 4.3. We call a compact quantum group G_{A} simple if it satisfies the following conditions (1)-(4):

(1) The Woronowicz C^{∗}-algebra A_{G} is finitely generated.

(2) G_{A} is connected.

(3) G_{A} has no non-trivial connected normal quantum subgroups.

(4) G_{A} has no non-trivial representations of dimension one.

A (simple) quantum group is called absolutely simple if it has no non-trivial normal quantum subgroups. Similarly a finite quantum group is called simple if it has no non-trivial normal quantum subgroups.

Just as the notion of simple compact Lie groups excludes the torus groups, this definition of simple quantum groups excludes the compact quantum groups coming from group C^{∗}-algebras C^{∗}(Γ) of discrete groups Γ (note that C^{∗}(Γ) is the algebra of continuous functions on the torus 𝕋^{n} when Γ is the abelian group ℤ^{n}). This is important because it is impossible to have a nice classification theory for discrete groups. However, we do not know if condition (4) in Definition 4.3 (i.e., there is no non-trivial group-like elements) is superfluous, as it is for simple compact Lie groups.

As a justification of this definition, we have the following proposition, which follows from Proposition 4.2 above and Sect. 2 of [40].

Proposition 4.4. Let G_{A} be an ordinary compact Lie group (i.e. A_{G} is commutative). Then G_{A} is a simple Lie group in the ordinary sense if and only if G_{A} is simple in the sense above.

Note that a simple compact Lie group is not a direct product of proper connected subgroups. Also, a simple Lie group is not a semi-direct product. By Proposition 3.2 and Proposition 3.4 (or [41]), similar results hold for quantum groups:

Proposition 4.5. If G_{A} is a simple compact quantum group, then A_{G} is not a tensor product, nor a crossed product by a non-trivial discrete group.

To put in perspective the examples of simple compact quantum groups to be studied later, we introduce some properties for compact quantum groups. First we recall that the representation ring (also called the fusion ring) R(G) of a compact quantum group G is an ordered algebra over the integers ℤ with positive cone (or semiring, which is also a basis) R(G)_{+} := {χ_{u}} consisting of characters χ_{u} of irreducible representations u ∈ Ĝ of G, and structure constants c_{uv}^{w} ∈ ℕ ∪ {0} given by the rules

χ_{u}χ_{v} = ∑_{w∈Ĝ} c_{uv}^{w}χ_{w},

where the product χ_{u}χ_{v} is taken in the algebra A_{G}.

Definition 4.6. Let G be a compact quantum group. We say that G has property F if each Woronowicz C^{∗}-subalgebra of A_{G} is of the form A_{G/N} for some normal quantum subgroup N of G. We say that G has property F D if each quantum subgroup of G is normal.

We say that G is almost classical if its representation ring R(G) is order isomorphic to the representation ring of a compact group.

By Proposition 2.3, a compact group trivially has property F. We will give in §5 and §6 non-trivial simple compact quantum groups that are almost classical and have property F. Among compact quantum groups, simple compact quantum groups that are almost classical or have property F are closest to ordinary simple compact Lie groups in regard to noncommutative geometry.

By Proposition 2.4, as the dual of discrete group Γ, a compact quantum group of the form C^{∗}(Γ) has property F D. When Γ is finite, C^{∗}(Γ) is equal to the dual of the function algebra C(Γ). This explains the term F D.

A compact quantum group G is absolutely simple with property F if and only if every non-trivial representation v of G is faithful, i.e., C^{∗}(v_{ij}) = A_{G}, cf. [46].

By a theorem of Handelman [16], the representation ring of a compact connected Lie group is a complete isomorphism invariant. But this fails for compact quantum groups, since the representation rings of a simple compact Lie group K and its standard deformation K_{q} are order isomorphic.

In [5], Banica uses the positive cone R_{+}(G) of the representation ring R(G) of a compact quantum group G to define what he calls an R_{+} deformation. This is closely related to almost classical quantum groups.

It is clear that a quantum quotient group G/N of an almost classical quantum group G is also almost classical. But a quantum subgroup of an almost classical quantum group need not be almost classical. For example, the quantum permutation groups are almost classical (cf. [4, 45] and remarks preceding Theorem 5.5), but according to of Bichon [8], their quantum subgroups A_{2}(ℤ/mℤ) are not almost classical if m ≥ 3 (see Corollary 2.7 and the paragraph following Corollary 4.3 of [8]). However, for a compact quantum group with property F (resp. F D), we have the following result. Theorem 4.7. Let G be a compact quantum group with property F (resp. F D). Then its quantum subgroups and quotient groups G/N (by normal quantum subgroups N) also have property F (resp. F D).

The assertions in the theorem above will not be needed in this paper. The details of their proofs will appear in [50].

The main goals in the theory of simple compact quantum groups are: (1) to classify up to isomorphism simple quantum groups (we must first find all of them) and (2) their irreducible representations; (3) to analyze the structure of compact quantum groups in terms of simple quantum groups; and (4) to develop applications of simple compact quantum groups in other areas of mathematics and physics. For these purposes, new techniques for compact quantum groups must be developed.

Before solving the classification problem, one should solve the following conceivably easier problems. Problem 4.8. Classify simple compact quantum groups with property F. Problem 4.9. Construct simple compact quantum groups with property F D. Problem 4.10. Construct simple compact quantum groups that are not almost classical.

5. Simplicity of B_{u}(Q) and A_{aut}(B, τ)

We recall the construction of compact quantum group B_{u}(Q) associated to a non-singular n × n complex scalar matrix Q (cf. [2, 40, 38, 44]). The (noncommutative) C^{∗}-algebra of functions on the quantum group B_{u}(Q) is generated by noncommutative coordinate functions u_{ij} (i, j = 1, ⋯, n) that are subject to the following relations:

u^{∗}u = I_{n} = uu^{∗}, u^{t}QuQ^{−1} = I_{n} = QuQ^{−1}u^{t},

where u = (u_{ij})_{i,j=1}^{n}. When QQ̄ is a scalar multiple cI_{n} of the identity matrix I_{n}, the quantum group B_{u}(Q) and the group SU(2) have the same fusion rules for their irreducible representations, as shown by Banica [2], which implies that B_{u}(Q) is an almost classical quantum group. Under the same condition on Q, the isomorphism classification of B_{u}(Q) is determined by the author in [45]. For arbitrary Q, B_{u}(Q) is shown also in [45] to be a free product of its building blocks, involving both B_{u}(Q)’s with QQ̄ being scalars and A_{u}(Q) with Q being positive (see also later in this section for A_{u}(Q)). For n = 1, B_{u}(Q) = C(𝕋) is the trivial 1 × 1 unitary group. We will concentrate on the non-trivial case n ≥ 2. As is shown in [45], the isomorphism class of B_{u}(Q) depends on the normalized Q only. Hence we assume Q satisfies the normalization condition QQ̄ = ±I_{n}. Theorem 5.1. Let Q ∈ GL(n, ℂ) be such that QQ̄ = ±I_{n}. Then B_{u}(Q) is a simple compact quantum group with property F. In fact it has only one normal subgroup of order 2.

Proof. Recall that according to Banica [2] irreducible representations of the quantum group B_{u}(Q) can be parametrized by r_{k} (k = 0, 1, 2, ⋯) with r_{0} trivial and r_{1} = (u_{ij})_{ij=1}^{n}, so that the fusion rules for their tensor product representations (i.e., decomposition into irreducible representations) read

r_{k} ⊗ r_{l} = r_{|k−l|} ⊕ r_{|k−l|+2} ⊕ ⋯ ⊕ r_{k+l−2} ⊕ r_{k+l}, k, l ≥ 0.

We first show that the quantum group B_{u}(Q) is connected. If k = 2m is even (m > 0), then let r_{l} = r_{k} in the above tensor product decomposition and do the same for the irreducible constituents repeatedly, one sees that the algebra C^{∗}(r_{2m}) generated by the coefficients of the representation r_{2m} contains the coefficients of r_{2s} for all s. Hence r_{2m} generates an infinite dimensional algebra:

C^{∗}(r_{2m}) = C^{∗}({r_{2s}|s ≥ 0}).

If k = 2m + 1 is odd (m ≥ 0), then let r_{l} = r_{k} in the above tensor product decomposition, one sees that the representation r_{2} appears therein. Apply the decomposition to r_{2m+1} ⊗ r_{2}, one sees that r_{1} = (u_{ij}) appears therein. Hence the algebra generated by the coefficients of r_{2m+1} is the same as the algebra generated by those of r_{1} = (u_{ij}). We conclude from this analysis that there is only one non-trivial Woronowicz C^{∗}-subalgebra in B_{u}(Q), the one C^{∗}(r_{2m}) generated by coefficients of r_{2m}, which is obviously infinite dimensional as noted above, where m is any nonzero positive number. In particular, the quantum group B_{u}(Q) is connected.

For rest of the proof, we show that the quantum group B_{u}(Q) has only one normal quantum subgroup, although it has many quantum subgroups.

Note that the coordinate functions v_{ij} of the matrix group N = {I_{n}, −I_{n}} satisfy the defining relations of B_{u}(Q), hence there is a surjection π from the C^{∗}-algebra B_{u}(Q) to the C^{∗}-algebra A_{N} of functions on N such that

π(u_{ij}) = v_{ij}, i, j = 1, 2, ⋯, n.

It is clear that π is a morphism of quantum groups, hence (N, π) is a quantum subgroup of the quantum group B_{u}(Q).

We show that (N, π) is actually a normal quantum subgroup. To see this, it suffices by Proposition 2.1 to show that

π(r_{2m}) = d_{2m} · v_{0}, π(r_{2m+1}) = d_{2m+1} · v_{1},

where d_{2m} and d_{2m+1} are dimensions of the representations r_{2m} and r_{2m+1} respectively, v_{0} and v_{1} are the trivial representation and the non-trivial irreducible representation of N respectively (v_{1}(±I_{n}) = ±1). By the definition of π and v_{1} the assertion is clearly true for m = 0. In general, suppose the assertion is true for m. Then π(r_{2m+1}) ⊗ π(r_{1}) is a multiple of v_{0} since v_{1}^{2} = v_{0}. From the decomposition of r_{2m+1} ⊗ r_{1}, we get

π(r_{2m+1}) ⊗ π(r_{1}) = π(r_{2m}) ⊕ π(r_{2m+2}).

Hence π(r_{2(m+1)}) = π(r_{2m+2}) is a scalar multiple of v_{0}. Similarly, from

π(r_{2m+2}) ⊗ π(r_{1}) = π(r_{2m+1}) ⊕ π(r_{2m+3}),

we see that π(r_{2(m+1)+1}) = π(r_{2m+3}) is a multiple of v_{1}. Since v_{0} and v_{1} are one dimensional representations, the multiples we obtained above must be d_{2m+2} and d_{2m+3} respectively. That is (N, π) is normal and

A_{G/N} = C^{∗}(r_{2}) = C^{∗}({r_{2s}|s ≥ 0}),

where for simplicity of notation, the symbol G in G/N refers to the quantum group G_{B_{u}(Q)}. The above also shows that this quantum group has property F.

We have to show that B_{u}(Q) has no other normal quantum subgroups, which will imply that it has no connected normal quantum subgroups and is therefore a simple quantum group.

Let (N_{1}, π_{1}) be a non-trivial normal quantum subgroup of B_{u}(Q). Since N_{1} ≠ 1, by Definition 2.7 and Proposition 2.1 there exists an irreducible representation v of the quantum group B_{u}(Q) such that π_{1}(v) is not a scalar and therefore E_{G/N_{1}}(v) = 0. Hence by the proof of Proposition 2.1 and Woronowicz’s Peter-Weyl theorem [52], A_{G/N_{1}} = E_{G/N_{1}}(A_{G}) ≠ A_{G}.

Similarly, we claim that A_{G/N_{1}} ≠ ℂ1, where 1 is the unit of A_{G}. To prove this, we need two lemmas. It is instructive to compare the first lemma with the ideal theory for C^{∗}-algebras. Lemma 5.2. (Hopf ∗-ideals vs. Woronowicz C^{∗}-ideals)

(1) Let G be a compact quantum group. Let I be a ∗-ideal of A_{G} satisfying

Δ(I) ⊂ A_{G} ⊗ I + I ⊗ A_{G} .

Then S(I) = I, ϵ(I) = 0, hence I is Hopf ∗-ideal of A_{G}. The norm closure \overline{I} in the C^{∗}-algebra A_{G} is a Woronowicz C^{∗}-ideal. The Hopf ∗-algebra A_{G}/I admits a universal C^{∗}-norm and its completion under this norm is a Woronowicz C^{∗}-algebra isomorphic to A_{G}/\overline{I}.

(2) The map f(I) = \overline{I} is a bijection from the set of Hopf ∗-ideals \{I\} of A_{G} onto the set of Woronowicz C^{∗}-ideals {I} of A_{G} with full A_{G}/I. The inverse g of f is given by g(I) = I ∩ A_{G}. Remarks. (a) Note that (2) and the last part of (1) in the lemma above are false if the Woronowicz C^{∗}-algebra A_{G} or A_{G}/I is not full, as is shown by the following example. Let A_{G} = C^{∗}(F_{2}) be the group C^{∗}-algebra of the free group F_{2} on two generators. Let I be the kernel of the canonical map π : C^{∗}(F_{2}) → C_{r}^{∗}(F_{2}) where C_{r}^{∗}(F_{2}) is the reduced group C^{∗}-algebra of F_{2}. Then I ∩ A_{G} = 0 but I ≠ \overline{0}.

(b) Part (1) of the lemma above strengths 2.10 of [40].

Proof of Lemma 5.2. Let π : A_{G} → A_{G}/\overline{I} be the quotient map. Then since

Δ(I) ⊂ A_{G} ⊗ I + I ⊗ A_{G} ⊂ ker(π ⊗ π),

we have that Δ(\overline{I}) ⊂ ker(π ⊗ π). That is, \overline{I} is a Woronowicz C^{∗}-ideal and therefore A_{G}/\overline{I} is a Woronowicz C^{∗}-algebra (see [40]). The dense Hopf ∗-subalgebra B_{0} of A_{G}/\overline{I} is the image under π of the Hopf ∗-algebra A_{G} and it is clearly isomorphic to A_{G}/I. As the kernel of this morphism π̂, I satisfies S(I) ⊂ I and ϵ(I) = 0, i.e. I is a Hopf ∗-ideal. The stronger assertion S(I) = I follows immediately from the property S(S(a^{∗})^{∗}) = a [52] (cf. 1.5.1 of [24] for a consequence of this assertion).

For an alternative (algebraic) argument to the one in the last paragraph, one can show that A_{G}/I is a compact quantum group algebra (CQG algebra) in the sense of Dijkhuizen and Koornwinder [13] (See also [19] and [53, 42]). But a better name for compact quantum group (CQG) algebra might be Woronowicz ∗-algebra (or compact Hopf ∗-algebra), since the quantum group C^{∗}-algebra of a compact quantum group G is the C^{∗}-algebra C^{∗}(G) dual to C(G) according to [26, 15].

Let B be the closure of A_{G}/I in the universal C^{∗}-norm. Then B is a Woronowicz C^{∗}-algebra. The composition

A_{G} → A_{G}/I → B

is bounded and extends to a morphism of Woronowicz C^{∗}-algebras from A_{G} to B. Let ρ denotes this morphism from A_{G} onto B. Since \overline{I} ⊂ ker(ρ), ρ factors through A_{G}/\overline{I} via a C^{∗}-algebra morphism ψ:

A_{G} ⟶^{π} A_{G}/\overline{I} ⟶^{ψ} B.

One can check that ψ is a morphism of Woronowicz C^{∗}-algebras [40]. Then the restriction ψ̂ of ψ to a morphism from the dense Hopf ∗-subalgebra B_{0} of A_{G}/\overline{I} to that of B is easily seen to be an isomorphism. Since B is full, ψ is itself an isomorphism from A_{G}/\overline{I} to B, see [42]. (We note in passing that since A_{G}/ ker(ρ) ≅ B, we have \overline{I} = ker(ρ).) This proves part of (1) of the lemma.

Let I be as in (2). Let B_{1} be the algebra A_{G}/\overline{I}. Then by (1) and [40] B_{1} is a Woronowicz C^{∗}-algebra. Let B_{1} be the dense Hopf ∗-algebra of B_{1}. Let π̂ : A_{G} → B_{1} be the morphism associated with the quotient morphism π. Then clearly

I ⊂ \overline{I} ∩ A_{G} = gf(I).

Conversely if x ∈ \overline{I} ∩ A_{G}, then x ∈ ker(π̂) = I. Hence gf(I) = I.

Let I be as in (2). We show that fg(I) = I. Let B_{2} = A_{G}/I and let π_{2} be the quotient morphism from A_{G} onto B_{2}. Define I = g(I) = I ∩ A_{G}. We need to show that \overline{I} = I. The idea of proof is the same as that of the last part in (1).

Using the morphism π̂_{2} : A_{G} → B_{2} associated with π_{2}, we see that I = ker(π̂_{2}). So I is a Hopf ∗-ideal in A_{G}. It follows from (1) that A_{G}/\overline{I} is a Woronowicz C^{∗}- algebra. Since \overline{I} ⊂ I, the morphism π_{2} factors through A_{G}/\overline{I} via a morphism ϕ of Woronowicz C^{∗}-algebras:

A_{G} ⟶^{π} A_{G}/\overline{I} ⟶^{ϕ} B_{2}.

The restriction ϕ̂ of ϕ to the dense Hopf ∗-algebras of A_{G}/\overline{I} to that of B_{2} becomes an isomorphism. Since B_{2} is full, ϕ is an isomorphism, which means that I = \overline{I} (and B_{1} = B_{2}). This completes the proof of Lemma 5.2.

Lemma 5.3. (Reconstruct N from G/N)

Let (N, π) be a normal quantum subgroup of a compact quantum group G. Let π̂ be the associated morphism from A_{G} to A_{N}. Then,

ker(π̂) = A_{G/N}^{+}A_{G} = A_{G}A_{G/N}^{+} = A_{G}A_{G/N}^{+}A_{G},

where H^{+} denotes the augmentation ideal (i.e. kernel of the counit) for any Hopf algebra H. Remarks. (a) In the notation of Schneider [32], the result above can be restated as follows: The map Φ is the left inverse of Ψ, where Ψ(ker(π̂)) := A_{G/N} and Φ(A_{G/N}) := A_{G}A_{G/N}^{+}. In the language of Andruskiewitsch et al [1], the result above implies that the sequence

1 ⟶ N ⟶ G ⟶ G/N ⟶ 1,

or the sequence

0 ⟶ A_{G/N} ⟶ A_{G} ⟶ A_{N} ⟶ 0,

is exact. It is instructive to compare this with the purely algebraic situation in [24], where for a given a normal quantum subgroup in their sense the existence of an exact sequence is not known and the uniqueness does not hold in general (cf. 1.6 and 6.3 loc. cit.). Note that the notion of exact sequence of quantum groups in [32] is equivalent to that in [1] under certain faithful (co)flat conditions, which is still not known to be valid in general. However, we have the following

Conjecture: Let G be a compact quantum group. Then the Hopf algebra A_{G} is faithfully flat over its Hopf subalgebras. An arbitrary Hopf algebra need not be faithfully flat over its Hopf subalgebras.

Similarly, one can formulate a conjecture concerning faithful coflatness.

(b) It follows from Lemma 5.3 and [32] that the notion of normal quantum groups in this paper and the one in [24] are equivalent for compact quantum groups.

Proof of Lemma 5.3. The proof is essentially the same as those of 16.0.2 in Sweedler [35] and (4.21) in Childs [10] for finite dimensional Hopf algebras, though Hopf algebras considered here are infinite dimensional in general. We sketch the main steps here for convenience of the reader.

It suffices to prove ker(π̂) = A_{G/N}^{+}A_{G}. The other equality ker(π̂) = A_{G}A_{G/N}^{+} is proved similarly. From these it follows that ker(π̂) = A_{G}A_{G/N}^{+}A_{G}.

Consider the right A_{N}-comodule structures on A_{N} and A_{G} given respectively by

Δ_{N} : A_{N} → A_{N} ⊗ A_{N}, and (id ⊗ π̂)Δ_{G} : A_{G} → A_{G} ⊗ A_{N},

where Δ_{N} and Δ_{G} are respectively the coproducts of the Hopf algebras A_{N} and A_{G}. Since A_{N} is cosemisimple by the fundamental work of Woronowicz [52, 40], it follows from Theorem 3.1.5 of [12] that every A_{N}-comodule is projective. Furthermore, one checks that the surjection π̂ : A_{G} → A_{N} is a morphism of A_{N}-comodules. Hence π̂ has a comodule splitting s : A_{N} → A_{G} with π̂s = id_{A_{N}}.

Let x ∈ A_{G/N}^{+}. By remark (a) following Definition 2.2, π̂(x) = 0. Hence A_{G/N}^{+} ⊂ ker(π̂) and therefore A_{G/N}^{+}A_{G} ⊂ ker(π̂).

Define a linear map ϕ on A_{G} by ϕ = (sπ̂) ∗ S = m(sπ̂ ⊗ S)Δ_{G}, where m and S are respectively the multiplication map and antipode of A_{G}. Then using the coassociativity of Δ_{G} and π̂s = id_{A_{N}} along with the antipodal property of S, one verifies that ϕ(A_{G}) ⊂ A_{G/N}. Since ker(π̂) ⊂ Im(id − sπ̂), to show ker(π̂) ⊂ A_{G/N}^{+}A_{G}, it suffices to show that Im(id − sπ̂) ⊂ A_{G/N}^{+}A_{G}. Since (ϵ − id)ϕ(A_{G}) ⊂ A_{G/N}^{+}, the later follows from the identity

id − sπ̂ = (ϵ − id)ϕ ∗ id = m((ϵ − id)ϕ ⊗ id)Δ_{G},

which one verifies using basic properties of the convolution product along with ϵϕ = ϵ and the splitting property of s. This proves Lemma 5.3.

Now we finish the proof of Theorem 5.1. If A_{G/N_{1}} = ℂ1, we would have A_{G/N_{1}} = ℂ1 and A_{G/N_{1}}^{+} = 0. Let π̂_{1} be the morphism of Hopf algebras from A_{G} to A_{N_{1}} associated with π_{1}. Then by Lemma 5.3,

ker(π̂_{1}) = A_{G/N_{1}}^{+}A_{G} = 0.

Since ker(π̂_{1}) is dense in ker(π_{1}) by Lemma 5.2, we would have ker(π_{1}) = 0. This contradicts the assumption that N_{1} is a non-trivial quantum subgroup of G and therefore A_{G/N_{1}} ≠ ℂ1. Then A_{G/N_{1}} has to be the only non-trivial Woronowicz C^{∗}- subalgebra of B_{u}(Q), i.e. A_{G/N_{1}} = C^{∗}({r_{2m}|m ≥ 0}) as noted near the beginning of the proof of Theorem 5.1. We infer from Proposition 2.1 that π_{1}(r_{2m}) is a multiple of the trivial representation of N_{1} for any m. From

π_{1}(r_{1}) ⊗ π_{1}(r_{1}) = π_{1}(r_{0}) ⊕ π_{1}(r_{2}),

we see that π_{1}(r_{1}) ⊗ π_{1}(r_{1}) is a multiple of the trivial representation of N_{1}. That is

∑_{ijkl} e_{ij} ⊗ e_{kl} ⊗ ũ_{ij}ũ_{kl} = I_{n} ⊗ I_{n} ⊗ 1,

where ũ_{ij} are the (i, j)-entries of π_{1}(r_{1}) and e_{ij} are matrix units. Hence

ũ_{ij}ũ_{kl} = 0, when i ≠ j, or k ≠ l;
ũ_{ii}ũ_{ll} = 1, for all i, l.

Therefore ũ_{ij} = 0 for i ≠ j and A_{N_{1}} is commutative. That is, N_{1} is an ordinary compact group. Now it is clear that ũ_{ii} = ũ_{ll} = ũ_{ll}^{−1} for all i, l, which we denote by a. Since A_{N_{1}} is generated by a and N_{1} is non-trivial, we conclude that N_{1} is a group of order 2. The map α from A_{N} to A_{N_{1}} defined by α(v_{ij}) = ũ_{ij} is clearly an isomorphism such that π_{1} = απ. Hence (N_{1}, π_{1}) = (N, π) by Definition 2.7.

For an example of a non-normal quantum subgroup (H, θ) of B_{u}(Q), take a two-elements group H = {I_{n}, V }, where

V = [ \begin{matrix}−1 & 0 \\ 0 & I_{n−1} \end{matrix} ] ,

and θ(u_{ij}) = w_{ij}, the coordinate functions on H. □

Let us also recall the construction of the quantum groups A_{u}(Q) closely related to B_{u}(Q) [40, 38, 44]. For every non-singular matrix Q, the quantum group A_{u}(Q) is defined in terms of generators u_{ij} (i, j = 1, ⋯ n), and relations:

u^{∗}u = I_{n} = uu^{∗}, u^{t}QūQ^{−1} = I_{n} = QūQ^{−1}u^{t}.

According to Banica [3], when Q > 0, the irreducible representations of the quantum group A_{u}(Q) are parameterized by the free monoid ℕ ∗ ℕ with generators α and β and anti-multiplicative involution ᾱ = β (the neutral element is e with ē = e). The classes of u and ū are r_{α} and r_{β} respectively. Moreover, for each pair of irreducible representations r_{x} and r_{y} (x, y ∈ ℕ ∗ ℕ), one has the following direct sum decomposition (fusion) rules:

r_{x} ⊗ r_{y} = ∑_{x=ag,ḡb=y} r_{ab}.

In [48], the special A_{u}(Q)’s with Q > 0 are classified up to isomorphism and the general A_{u}(Q)’s with arbitrary Q are shown to be free products of the special A_{u}(Q)’s. However, the following problem is open. Problem 5.4. Determine whether the quantum groups A_{u}(Q) are simple for positive Q ∈ GL(n, ℂ) and n ≥ 2.

To prove the simplicity of A_{u}(Q), one would only need to show that A_{u}(Q) has no non-trivial connected normal quantum subgroups, since the other conditions are easily verified. If this is indeed true, A_{u}(Q) will provide the first examples of simple compact quantum groups that are not almost classical.

Now we consider the quantum automorphism group A_{aut}(B, tr) of a finite dimensional C^{∗}-algebra B endowed with a tracial functional tr (cf. [4, 45]). This quantum group is defined to be the universal object in the category of compact quantum transformation groups of B that leave tr invariant. Note that the presence of a tracial functional tr is necessary for the existence of the universal object when B is non-commutative (see Theorem 6.1 of [45]). For an arbitrary finite dimensional C^{∗}-algebra B, the C^{∗}-algebra A_{aut}(B, tr) is described explicitly in [45] in terms of generators and relations. When B = C(X_{n}) is the commutative C^{∗}- algebra of functions on the space X_{n} of n points, the quantum automorphism group A_{aut}(B) = A_{aut}(X_{n}) (also called the quantum permutation group on n letters) exists without the presence of a (tracial) functional and its description in terms of generators and relations is surprisingly simple. The C^{∗}-algebra A_{aut}(X_{n}) is generated by self-adjoint projections a_{ij} such that each row and column of the matrix (a_{ij})_{i,j=1}^{n} add up to 1. That is,

a_{ij}^{2} = a_{ij} = a_{ij}^{∗}, i, j = 1, ⋯, n,

∑_{j=1}^{n} a_{ij} = 1, i = 1, ⋯, n,

∑_{i=1}^{n} a_{ij} = 1, j = 1, ⋯, n.

For more general finite dimensional C^{∗}-algebras B, the description of A_{aut}(B, tr) in terms of generators and relations is more complicated. We refer the reader to [45] for detailed description.

Assume tr is the canonical trace τ on B (see §1 or p772 of [4] for the definition). Then A_{aut}(B, τ) is an ordinary permutation group when the dimension of B is less or equal to 3. However, when the dimension of B is greater or equal to 4, A_{aut}(B, τ) is a non-trivial (noncommutative and noncocommutative) compact quantum group with an infinite dimensional function algebra [45, 46]. Moreover, the representation ring of A_{aut}(B, τ) is isomorphic to that of SO(3) [4]. Hence A_{aut}(B, τ) is almost classical for all B. It is easy to see that for B = C(X_{n}), the canonical trace τ is equal to the unique S_{n}-invariant state on B, where S_{n} acts on X_{n} by permutation. Hence by remark (2) following Theorem 3.1 of [45], A_{aut}(B, τ) is the same as the quantum permutation group A_{aut}(X_{n}).

We refer the reader to [4, 45, 46] for more on these quantum groups and [7, 8] for related results. Note that the description in [4] is not exactly as that in [45] but equivalent to it. We now prove

Theorem 5.5. Let B be a finite dimensional C^{∗}-algebra with dim(B) ≥ 4. Then A_{aut}(B, τ) is an absolutely simple compact quantum group with property F.

Proof. The argument is similar to that of Theorem 5.1. First we claim that there are only two Woronowicz C^{∗}-subalgebras in A_{aut}(B, τ), namely ℂ1 and A_{aut}(B, τ).

By Banica [4], the complete set of mutually inequivalent irreducible representations of the quantum group A_{aut}(B, τ) can be parametrized by r_{k} (k ≥ 0, r_{0} being the trivial one dimensional representation). Under this parametrization the fusion rules of the irreducible representations are the same as those of SO(3):

r_{k} ⊗ r_{l} = r_{|k−l|} ⊕ r_{|k−l|+1} ⊕ ⋯ ⊕ r_{k+l−1} ⊕ r_{k+l}, k, l ≥ 0.

Let A_{1} ≠ ℂ1 be a Woronowicz C^{∗}-subalgebra of A_{aut}(B, τ). Let v be a nontrivial irreducible representation of the compact quantum group of A_{1}. Then v = r_{k} for some k ≠ 0 and

r_{k} ⊗ r_{k} = r_{0} ⊕ r_{1} ⊕ r_{2} ⊕ ⋯ ⊕ r_{2k−1} ⊕ r_{2k}, k, l ≥ 0.

Hence the coefficients of each of the representations r_{1}, r_{2}, ..., r_{2k} are in A_{1}. Similarly, from the decomposition of r_{2k} ⊗ r_{2k}, we see that the coefficients of each of the representations r_{1}, r_{2}, ..., r_{4k} are in A_{1}. Inductively, the coefficients of each of the representations r_{1}, r_{2}, ..., r_{2^{m}k} are in A_{1} (m > 0). Hence A_{1} = A_{aut}(B, τ).

Let (π, N) be a normal quantum subgroup of G = A_{aut}(B, τ) different from the trivial one-element subgroup. Then there is a non-trivial irreducible representation u^{λ} = (u_{ij}^{λ}) such that π(u^{λ}) is not a multiple of the trivial representation. Using the same argument as in the proof of Theorem 5.1 we have A_{G/N} = E_{G/N}(A_{G}) ≠ A_{G}. Therefore we must have A_{G/N} = ℂ1. Then the argument near the end of the proof of Theorem 5.1 (i.e. the paragraph that follows the proof of Lemma 5.3) shows that ker(π) = 0. That is, N is the same quantum group as G. □

Theorem 5.5 applies in particular to quantum permutation groups A_{aut}(X_{n}) when n ≥ 4. As Manin pointed out (private communication), the reason that these quantum groups are connected could be that there are so many more quantum symmetries that the originally n! isolated permutations are connected together by them. Note however that their function algebras are generated by orthogonal projections a_{ij}, so these quantum groups are also disconnected, as observed by Bichon [8] recently. It would be interesting to find a satisfactory explanation of this quantum paradox.

The proofs of the main results of this section does not need explicit description (models) of representations of the quantum groups B_{u}(Q) and A_{aut}(B, τ). Only the structures of their representation rings (fusion rules) are used. However, explicit constructions of models of irreducible representations of Lie groups are more interesting and have more applications than the structures of their representation rings. In view of this, we believe an answer to the following problem will be even more useful in the theory of quantum groups. (Note that the model for the fundamental representation of the quantum group A_{u}(Q) is used in [46] to construct ergodic actions on various von Neumann factors.)

Problem 5.6. Construct explicit models of the irreducible representations of the following quantum groups: A_{u}(Q) for Q > 0; B_{u}(Q) for QQ̄ = ±I_{n}; the quantum automorphism group A_{aut}(B, τ) of a finite dimensional C^{∗}-algebra B endowed with the canonical trace τ.

6. Simplicity of K_{q}, K_{q}^{u} and K_{J}

The compact real forms K_{q} of Drinfeld-Jimbo quantum groups and their twists K_{q}^{u} are studied in [33] and [21] respectively. See also [22] for a summary of [33, 21] and [20] for more detailed treatment. Motivated by these works, Rieffel constructs in [28] a deformation K_{J} of compact Lie group K which contains a torus T and raises the question whether K_{q}^{u} can be obtained as a strict deformation quantization of K_{q}. This question is answered in the affirmative by the author in [43]. The purpose of this section is to show that the quantum groups K_{q}, K_{q}^{u} and K_{J} are simple in the sense of this paper, provided that the compact Lie group K is simple.

We first recall the notation of [33, 21, 22]. Let G be a connected and simply connected simple complex Lie group with Lie algebra 𝔤. Fix a triangular decomposition 𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+}, together with the corresponding decomposition Δ = Δ_{+} ∪ Δ_{−} of the root system and a fixed basis {α_{i}}_{i=1}^{n} for Δ_{+}. For each linear functional λ on 𝔥, H_{λ} denotes the element in 𝔥 corresponding to λ under the isomorphism 𝔥 ≅ 𝔥^{∗} determined by the Killing form ( , ) on 𝔤. Note that if the reader keeps the context in mind, the symbols α and λ used in this context should not cause confusion with the same symbols used in this paper for other purposes. Let {X_{α}}_{α∈Δ} ∪ {H_{i}}_{i=1}^{n} be a Weyl basis of 𝔤, where H_{i} = H_{α_{i}}. This determines a Cartan involution ω_{0} on 𝔤 with ω_{0}(X_{α}) = −X_{−α}, ω_{0}(H_{i}) = −H_{i}. Let 𝔨 be the compact real form of 𝔤 defined as the fixed points of ω_{0} and K the associated compact real form of G. Put 𝔥_{ℝ} = ⊕_{i=1}^{n}ℝH_{i}, 𝔱 = i𝔥_{ℝ} and T = exp(𝔱), the later being the associated maximal torus of K.

Let q = e^{h/4} (h ∈ ℝ\{0}). For n, k ∈ ℕ, n ≥ k, define

[n]_{q} = \frac{q^{n} − q^{−n}}{q − q^{−1}} ,
\left[{n \atop k}\right]_{q} = \frac{[n]_{q}[n − 1]_{q} … [n − k + 1]_{q}}{[k]_{q}[k − 1]_{q} … [1]_{q}} .

The quantized universal enveloping algebra U_{q}(𝔤) [14, 18] is the complex associative algebra with generators X_{i}^{±}, K_{i}^{±1} (i = 1, ⋯, n) and defining relations:

K_{i}K_{i}^{−1} = 1 = K_{i}^{−1}K_{i}, K_{i}K_{j} = K_{j}K_{i},

K_{i}X_{j}^{±}K_{i}^{−1} = q^{±(α_{i},α_{j})}X_{j}^{±},

[X_{i}^{+}, X_{j}^{−}] = δ_{ij} \frac{K_{i}^{2} − K_{i}^{−2}}{q − q^{−1}} ,

∑_{k=0}^{1−a_{ij}}(−1)^{k} \left[{1 − a_{ij} \atop k}\right]_{q_{i}} (X_{i}^{±})^{k}X_{j}^{±}(X_{i}^{±})^{1−a_{ij}−k} = 0 , i ≠ j,

where q_{i} = q^{(α_{i},α_{i})}.

On U_{q}(𝔤) there is a Hopf algebra structure with coproduct

Δ(K_{i}^{±1}) = K_{i}^{±1} ⊗ K_{i}^{±1}, Δ(X_{i}^{±}) = X_{i}^{±} ⊗ K_{i} + K_{i}^{−1} ⊗ X_{i}^{±},

and counit and antipode respectively

ε(X_{i}^{±}) = 0, ε(K_{i}^{±1}) = 1, S(X_{i}^{±}) = −q_{i}^{±1}X_{i}^{±}, S(K_{i}^{±1}) = K_{i}^{∓1}.

Under the *-structure defined by

(X_{i}^{±})^{∗} = X_{i}^{∓}, K_{i}^{∗} = K_{i},

U_{q}(𝔤) is a Hopf *-algebra.

Let u = ∑_{k,l} c_{kl}H_{k} ⊗ H_{l} ∈ ∧^{2}𝔥_{ℝ}. Then it can be shown (cf. [20]) that the following defines a new coproduct on U_{q}(𝔤),

Δ_{u}(ξ) = exp(−ihu/2)Δ(ξ) exp(ihu/2),

where X ∈ U_{q}(𝔤) and Δ is the original coproduct on U_{q}(𝔤). The new Hopf *-algebra so obtained is denoted by U_{q,u}(𝔤).

The function algebra A_{K_{q}} of the compact quantum group K_{q} is defined to be the subalgebra of the dual algebra U_{q}(𝔤)^{∗} consisting of matrix elements of finite dimensional representations ρ of U_{q}(𝔤) such that eigenvalues of the endomorphisms ρ(K_{i}) are positive. The function algebra A_{K_{q}^{u}} of the compact quantum group K_{q}^{u} is defined to be the subalgebra of the dual algebra U_{q,u}(𝔤)^{∗} that has the same elements as A_{K_{q}}, as well as the same ∗-structure, while the product of its elements is defined using Δ_{u} instead of Δ.

For each (algebraically) dominant integral weight λ ∈ P_{+} of (𝔤, 𝔥), define matrix elements C_{μ,i;ν,j}^{λ} of the highest weight U_{q}(𝔤) module (L(λ), ρ_{λ}) as follows. Let {v_{ν}^{(i)}} be an orthonormal weight basis for the unitary U_{q}(𝔤) module L(λ). Then C_{μ,i;ν,j}^{λ} is defined by

C_{μ,i;ν,j}^{λ}(X) =< ρ_{λ}(X)v_{ν}^{(j)}, v_{μ}^{(i)} >,

where X ∈ U_{q}(𝔤) and < , > is the inner product on L(λ). The C_{μ,i;ν,j}^{λ}’s is a linear (Peter-Weyl) basis of both A_{K_{q}} and A_{K_{q}^{u}} when λ ranges through the set P_{+} of dominant integral weights of (𝔤, 𝔥). By [23, 30] it is clear from these definitions that K_{q} and K_{q}^{u} are almost classical quantum groups in the sense of Definition 4.6. Theorem 6.1. Let K be a connected and simply connected simple compact Lie group. Then K_{q} is a simple compact quantum group with property F.

Proof. First we recall that representations of K and K_{q} are in one to one correspondence via deformation and the decompositions of tensor products of irreducible representations are not altered under deformation (see Lusztig [23] and Rosso [30] or Chari-Pressley [9]). Let ξ be the map that associates each irreducible representation v of K an irreducible representation ξ(v) of K_{q} in this correspondence. This map defines an isomorphism of vector spaces from A_{K} to A_{K_{q}}, which we also denote by ξ. It follows from this that K_{q} is connected and has no non-trivial representations of dimension one. Comparing decompositions of tensor products of representations of K and K_{q} we see that the ξ maps bijectively the set of Hopf subalgebras of A_{K} onto the set of Hopf subalgebras of A_{K_{q}}.

Let ρ_{q} be the quotient morphism from A_{K_{q}} to the abelianization A_{K_{q}}^{ab}, which is by definition the quotient of A_{K_{q}} by the closed two sided ideal of A_{K_{q}} generated by commutators [a, b], a, b ∈ A_{K_{q}}. According to [40], A_{K_{q}}^{ab} is the algebra of continuous functions on the maximal compact subgroup Â_{K_{q}} of K_{q} and ρ_{q} gives rise to the embedding of the quantum groups from Â_{K_{q}} to K_{q}. It is shown in [33] that the maximal compact subgroup Â_{K_{q}} is isomorphic to the maximal torus T of K.

The associated morphism ρ̂_{q} from A_{K_{q}} to A_{T} is given by

ρ̂_{q}(C_{μ,i;ν,j}^{λ})(t) = δ_{ij}δ_{μν}e^{2πμ(x)},

where t = exp(x) ∈ T, x ∈ 𝔱 = i𝔥_{ℝ} (see p438 of [9], but \sqrt{−1} should not appear in the formula there). It is clear that one has the same formula as above for restriction morphism ρ from A_{K} to A_{T}:

ρ̂(ξ^{−1}(C_{μ,i;ν,j}^{λ}))(t) = δ_{ij}δ_{μν}e^{2πμ(x)}, i.e., ρ̂ = ρ̂_{q} ∘ ξ.

Let N ⊂ K be a normal subgroup of K with surjections π : A_{K} → A_{N} and π̂ : A_{K} → A_{N}. Then N is a finite subgroup of T and A_{N} = A_{N} is a finite dimensional Hopf algebra. It is clear that π = ρ_{N} ∘ ρ, where ρ_{N} is the restriction morphism from A_{T} to A_{N}. Define

π_{q} : A_{K_{q}} ⟶ A_{N}, by π_{q} := ρ_{N} ∘ ρ_{q}.

We claim that (N, π_{q}) is a normal subgroup of K_{q}. This follows immediately from the following identities, which one can easily verify using ρ̂ = ρ̂_{q} ∘ ξ and π̂ = π̂_{q} ∘ ξ.

A_{K_{q}/N} = ξ(A_{K/N}), i.e.,
{a ∈ A_{K_{q}}|(id ⊗ π_{q})Δ(a) = a ⊗ 1} = ξ({a ∈ A_{K}|(id ⊗ π)Δ(a) = a ⊗ 1});
A_{N\K_{q}} = ξ(A_{N\K}), i.e.,
{a ∈ A_{K_{q}}|(π_{q} ⊗ id)Δ(a) = 1 ⊗ a} = ξ({a ∈ A_{K}|(π ⊗ id)Δ(a) = 1 ⊗ a}).

That is, every normal subgroup N of K gives rise to a normal subgroup (N, π_{q}) of K_{q} in the manner above.

Conversely, let (N′, π′) be a quantum normal subgroup of K_{q}. Then A_{K_{q}/N′} is a Hopf subalgebra of A_{K_{q}}. Since every Hopf subalgebra of A_{K} is of the form A_{K/N} for some normal subgroup N of K (cf. [40]), by the correspondence between Hopf subalgebras of A_{K} and those of A_{K_{q}} noted near the beginning of the proof we have

A_{K_{q}/N′} = ξ(A_{K/N}) = A_{K_{q}/N}

for some normal subgroup N of K. By Lemma 5.3, we have ker(π̂) = ker(π̂′). That is (N′, π′) and (N, π_{q}) are the same quantum subgroup of K_{q} (cf. Definition 2.7 and Lemma 5.2). Since normal subgroups N of K are finite, we conclude from the above that K_{q} has no non-trivial connected quantum normal subgroups. □

Examining the proof of Theorem 6.1, we formulate the following general result on the invariance of simplicity of compact quantum groups under deformation, which will be used to prove the simplicity of K_{q}^{u} and K_{J}.

Let G be a simple compact quantum group with property F and (H, ρ) a quantum subgroup. Assume all normal quantum subgroups of G are quantum subgroups of H. Let G_{v} be a family of compact quantum groups (“deformation” of G) indexed by a subset {v} of a vector space that includes the origin. Suppose the family G_{v} satisfy the following conditions:

C1. G_{0} = G.

C2. There is an isomorphism ξ of vector spaces from A_{G} to A_{G_{v}}.

C3. The coproduct is unchanged under deformation, i.e.,

Δ_{v}(ξ(a)) = (ξ ⊗ ξ)Δ(a) for a ∈ A_{G}.

C4. For any pair irreducible representations u^{λ_{1}} and u^{λ_{2}}, if

u^{λ_{1}} ⊗ u^{λ_{2}} ≅ u^{γ_{1}} ⊕ u^{γ_{2}} ⊕ ⋯ ⊕ u^{γ_{l}}

is a decomposition of u^{λ_{1}} ⊗ u^{λ_{2}} into direct sum of irreducible subrepresentations u^{γ_{j}} (j = 1, 2, ⋯, l), then

ξ(u^{λ_{1}}) ⊗ ξ(u^{λ_{2}}) ≅ ξ(u^{γ_{1}}) ⊕ ξ(u^{γ_{2}}) ⊕ ⋯ ⊕ ξ(u^{γ_{l}})

is a decomposition of ξ(u^{λ_{1}}) ⊗ ξ(u^{λ_{2}}) into direct sum of irreducible representations, where for instance ξ(u^{λ_{1}}) denotes the representation of G_{v} whose coefficients are images of coefficients of u^{λ_{1}}.

C5. The quantum subgroup H is undeformed. The latter means that there is a morphism ρ_{v} of quantum groups from H to G_{v} such that

ρ_{v}(ξ(a)) = ρ(a) for a ∈ A_{G}.

Under the assumptions above, we have the following result. The proof is the same as that of Theorem 6.1 (H corresponds to T in Theorem 6.1). Theorem 6.2. The compact quantum groups G_{v} are simple with property F. Remarks. (a) Condition C4 above is not the same as the requirement that

ξ(u^{λ_{1}} ⊗ u^{λ_{2}}) = ξ(u^{λ_{1}}) ⊗ ξ(u^{λ_{2}}).

The latter requirement together with conditions (2) and (3) imply that ξ is an isomorphism of quantum from G_{v} to G, which is not the case for the quantum groups under consideration here.

(b) We believe similar results on invariance of simplicity under deformation hold true without the property F assumption on G. But at the moment we do not know of any simple compact quantum groups that do not satisfy this property, though there many non-simple quantum groups without this property.

Next we recall the construction in [28, 43]. Let A = A_{G} be a compact quantum group with coproduct Δ. Suppose that the quantum group G has a toral subgroup (T, ρ)–to obtain non-trivial deformation we assume that T has rank no less than 2. For any element t in T, denote by E_{t} the corresponding evaluation functional on C(T). Assume that η is a continuous homomorphism from a vector space Lie group ℝ^{n} to T, where n is allowed to be different from the dimension of T. Define an action α of ℝ^{d} := ℝ^{n} × ℝ^{n} on the C^{∗}-algebra A as follows:

α_{(s,v)} = l_{η(s)}r_{η(v)},

where

l_{η(s)} = (E_{η(−s)}ρ ⊗ id)Δ, r_{η(v)} = (id ⊗ E_{η(v)}ρ)Δ.

For any skew-symmetric operator S on ℝ^{n}, one may apply Rieffel’s quantization procedure [27] for the action α above to obtain a deformed C^{∗}-algebra A_{J} whose product is denoted ×_{J}, where J = S ⊕ (−S). The family A_{hJ} (h ∈ ℝ) is a strict deformation quantization of A (see Chapter 9 of [27]). In [43] the following result is obtained. Theorem 6.3. The deformation A_{J} is a compact quantum group containing T as a (quantum) subgroup; A_{J} is a compact matrix quantum group if and only if A is.

We denote by G_{J} the quantum group for A_{J}. When G is a compact Lie group, the construction G_{J} above is the same as Rieffel’s construction [28]. By 5.2 of [28], G_{J} is an almost classical compact quantum group if G is a compact Lie group.

Combining Theorem 6.3 and Theorem 6.2, we obtain Theorem 6.4. Let K be a simple compact Lie group with a toral subgroup T. Then the compact quantum group K_{J} of Rieffel [28] is simple with property F.

We note that unlike in Theorem 6.1, in the result above we do not need to assume K to be simply connected. This is because A_{K_{q}} is defined using irreducible representations of U_{q}(𝔤) associated with all dominant integral weights P_{+} of (𝔤, 𝔥), so that A_{K_{q}} becomes the algebra of representative functions on a simply connected K when q → 1. One could also start with a non-simply connected K in Theorem 6.1 too, but then one needs to modify the definition of the quantum algebra A_{K_{q}} by using irreducible representations of U_{q}(𝔤) associated with analytically dominant integral weights only. This newly defined A_{K_{q}} is a Hopf subalgebra of the Hopf algebra defined originally. It is clear from the proof of Theorem 6.1 that its conclusion remains valid for this newly defined K_{q}.

Finally we consider K_{q}^{u}. To avoid confusion with the Killing form, we now use s ⊕ v, instead of (s, v) as is used above, to denote an element of ℝ^{d} = ℝ^{n} × ℝ^{n}. In the present setting, the space ℝ^{n} is 𝔥_{ℝ}, with inner product < , >= ( , ), where ( , ) is the Killing form of 𝔤 restricted to 𝔥_{ℝ}. We will also use < , > to denote the inner product on 𝔥_{ℝ} ⊕ 𝔥_{ℝ}. Noting that the compact abelian group T is also a subgroup of both K_{q} and K_{q}^{u} (see [33, 21]). The map η there in this case is defined by η(s) = exp(2πis). We can define as above an action of ℝ^{d} on A_{K_{q}} by

α_{s⊕v} = l_{exp(−2πis)}r_{exp(2πiv)}.

This action may be viewed as an action of H = T × T in the sense of [27]. For each ν in the weight lattice P of 𝔤, the element H_{ν} is in 𝔥_{ℝ}. We use the notation H_{ν} ⊕ H_{μ} to denote H_{ν} + H_{μ} as an element of 𝔥_{ℝ} ⊕ 𝔥_{ℝ}. Keep the notation of [27] for the spectral subspaces of the action α (see 2.22 there).

Let ǔ be the map on 𝔥* determined by u via the Killing form ( , ) on 𝔤. Let

p = −(H_{ν_{1}} ⊕ H_{μ_{1}}), q = −(H_{ν_{2}} ⊕ H_{μ_{2}}),

J = \frac{h}{4π} (S_{u} ⊕ (−S_{u})),

where S_{u} is the skew-symmetric operator on 𝔥_{ℝ} defined by

S_{u}(H_{ν}) = ∑_{k,l} c_{kl}ν(H_{k})H_{l}.

Then one has

C_{μ_{1},i_{1};ν_{1},j_{1}}^{λ_{1}} ∘ C_{μ_{2},i_{2};ν_{2},j_{2}}^{λ_{2}}

= exp(\frac{ih}{2}((μ_{1}, ǔμ_{2}) − (ν_{1}, ǔν_{2})))C_{μ_{1},i_{1};ν_{1},j_{1}}^{λ_{1}}C_{μ_{2},i_{2};ν_{2},j_{2}}^{λ_{2}}

= exp(−2πi < p, Jq >)C_{μ_{1},i_{1};ν_{1},j_{1}}^{λ_{1}}C_{μ_{2},i_{2};ν_{2},j_{2}}^{λ_{2}}

where ∘ on the left-hand side is the multiplication in A_{K_{q}^{u}} and the right-hand side is the multiplication in A_{K_{q}}.

On the other hand one has from 2.22 of [27] that

C_{μ_{1},i_{1};ν_{1},j_{1}}^{λ_{1}} ×_{J} C_{μ_{2},i_{2};ν_{2},j_{2}}^{λ_{2}} = exp(−2πi < p, Jq >)C_{μ_{1},i_{1};ν_{1},j_{1}}^{λ_{1}}C_{μ_{2},i_{2};ν_{2},j_{2}}^{λ_{2}}

This means that we have the following result [43]. Theorem 6.5. The Hopf *-algebras A_{K_{q}^{u}} and (A_{K_{q}}, ×_{J}) are isomorphic.

That is K_{q}^{u} = (K_{q})_{J} in the notation of Theorem 6.3, answering Rieffel’s question [28] in the affirmative.

Combining Theorem 6.5, Theorem 6.1 and Theorem 6.2, we obtain the following Theorem 6.6. Let K be a connected and simply connected simple compact Lie group. Then the compact quantum group K_{q}^{u} is simple with property F.

Recently, Connes and Dubois-Violette [11] found the so called θ-deformation SO_{θ}(n) of SO(n). It would be interesting to know if these new quantum groups are simple compact quantum groups also.

Acknowledgments. The research reported here is principally supported by the National Science Foundation grant DMS-0096136. During the writing of this work, the author also received support from the Research Foundation of the University of Georgia in the form of a Faculty Research Grant in the summer of 2000 and the Max Planck Institute of Mathematics at Bonn in the form of a visiting membership for 6 weeks in the summer of 2002. The author is grateful to these agencies and institutions for generous support.

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Department of Mathematics, University of Georgia, Athens, GA 30602 Fax: 706-542-2573

   E-mail address: szwang@math.uga.edu