Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

Let $\mathfrak{g}$ be a compact simple Lie algebra. We modify the quantized enveloping $^*$-algebra associated to $\mathfrak{g}$ by a real-valued character on the positive part of the root lattice. We study the ensuing Verma module theory, and the associated quotients of these modified quantized enveloping $^*$-algebras. Restricting to the locally finite part by means of a natural adjoint action, we obtain in particular examples of quantum homogeneous spaces in the operator algebraic setting.


Introduction
This paper reports on preliminary work related to the quantization of non-compact semi-simple Lie groups. The main idea behind such a quantization is based on the reflection technique developed in [5] and [11] (see also [7] and [6] for concrete, small-dimensional examples relevant to the topic of this paper). Briefly, this technique works as follows. Let G be a compact quantum group acting on a compact quantum homogeneous space X. Assume that the von Neumann algebra L ∞ (X) associated to X is a type I factor. Then the action of G on L ∞ (X) can be interpreted as a projective representation of G, and one can deform G with the 'obstruction' associated to this projective representation to form a new locally compact quantum group H. More generally, if L ∞ (X) is only a finite direct sum of type I-factors, one can construct H as a locally compact quantum groupoid (of a particularly simple type). Our idea is to fit the quantizations of non-compact semi-simple Lie groups into this framework, obtaining them as a reflection of the quantization of their compact companion. For this, one needs the proper quantum homogeneous spaces to feed the machinery with.
It is natural to expect the needed quantum homogeneous space to be a quantization of a compact symmetric space associated to the non-compact semi-simple Lie group. By now, there is much known on the quantization of symmetric spaces (see [24,25] and references therein, and [31] for the non-compact situation), but these results are mostly of an algebraic nature, and not much seems known about corresponding operator algebraic constructions except for special cases. In fact, in light of the motivational material presented in Appendix B, we will instead of symmetric spaces use certain quantizations of (co)adjoint orbits, following the approach of [10,19,28]. Here, one rather constructs quantum homogeneous spaces as subquotients of (quantized) This paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel. The full collection is available at http://www.emis.de/journals/SIGMA/Rieffel.html arXiv:1307.3642v3 [math.RT] 24 Dec 2013 universal enveloping algebras in certain highest weight representations. We will build on this approach by combining it with real structures and the contraction technique.
Our main result, Theorem 3.20, will consist in showing that the compact quantum homogeneous spaces that we build do indeed consist of finite direct sums of type I factors. This will give a theoretical underpinning and motivation for the claim that the above mentioned quantizations of non-compact semi-simple groups can indeed be constructed using the reflection technique. Our results are however quite incomplete as of yet, as • in the non-contracted case, we can only treat concretely the case of Hermitian symmetric spaces, • a more detailed analysis of the resulting quantum homogeneous spaces is missing, • the relation to known quantum homogeneous spaces is not elucidated, • no precise connection with deformation quantization is provided, • the relation with the approach of Korogodsky [23] towards the quantization of non-compact Lie groups remains to be clarified.
We hope to come back to the above points in future work. The structure of this paper is as follows. In Section 1, we introduce the 'modified' quantized universal enveloping algebras we will be studying, and state their main properties in analogy with the ordinary quantized universal enveloping algebras. In Section 2, we introduce a theory of Verma modules, and study the associated unitarization problem. In Section 3, we study subquotients of our generalized quantized universal enveloping algebras, and show how they give rise to C * -algebraic quantum homogeneous spaces whose associated von Neumann algebras are direct sums of type I factors. In Section 4, we briefly discuss a case where the associated von Neumann algebra is simply a type I-factor itself.
In the appendices, we give some further comments on the structures appearing in this paper. In Appendix A, we recall the notion of cogroupoids [2] which is very convenient for our purposes. In Appendix B, we discuss the Lie algebras which are implicitly behind the constructions in the main part of the paper.
1 Two-parameter deformations of quantized enveloping algebras Let g be a complex simple Lie algebra of rank l, with fixed Cartan subalgebra h and Cartan decomposition g = n − ⊕ h ⊕ n + . Let ∆ ⊆ h * be the associated finite root system, ∆ + the set of positive roots, and Φ + = {α r | r ∈ I} the set of simple positive roots. We identify I and Φ + with the set {1, . . . , l} whenever convenient. Let h * R ⊆ h * be the real linear span of the roots, and let ( , ) be an inner product on h * R for which A = (a rs ) r,s∈I = ((α ∨ r , α s )) r,s∈I is the Cartan matrix of g, where α ∨ = 2 (α,α) α for α ∈ ∆. Let h R ⊆ h be the real linear span of the coroots h α , where β(h α ) = (β, α ∨ ) for α, β ∈ ∆.
We further use the following notation. We write {ω r | r ∈ I} for the fundamental weights in h * R , so (ω r , α ∨ s ) = δ rs . The Z−lattice spanned by {ω r } is denoted P ⊆ h * R , and P + denotes elements expressed as positive linear combinations of this basis. Similarly, the root lattice spanned by the α r is denoted Q, and its positive span by Q + . We write Char K (F ) for the monoid of monoid homomorphisms from a commutative (additive) monoid (F, +) to a commutative (multiplicative) monoid (K, ·). For ε ∈ Char K (Q + ) we will abbreviate ε αr = ε r . The unit element of Char K (Q + ) will be denoted +, while the element ε such that ε r = 0 for all r will be denoted 0. We use the following notation for q-numbers, where 0 < q < 1 is fixed for the rest of the paper: • for α ∈ h * , we define q α ∈ Char C (P ) by q α (ω) = q (α,ω) (where ( , ) has been C-linearly extended to h).
We will only work with unital algebras defined over C, and correspondingly all tensor products are algebraic tensor products over C. By a * -algebra A will be meant an algebra A endowed with an anti-linear, anti-multiplicative involution * : A → A. We further assume that the reader is familiar with the theory of Hopf algebras. A Hopf * -algebra (H, ∆) is a Hopf algebra whose underlying algebra H is a * -algebra, and whose comultiplication ∆ is a * -homomorphism. This implies that the counit is a * -homomorphism, and that the antipode S is invertible with Definition 1.1. For ε, η ∈ Char R (Q + ), we define U q (g; ε, η) as the universal unital * -algebra generated by couples of elements X ± r , r ∈ I, as well as elements K ω , ω ∈ P , such that for all r, s ∈ I and ω, χ ∈ P , we have K ω self-adjoint, (X + r ) * = X − r and (K) q K ω is invertible and When ε = η = + (i.e. ε r = η r = 1 for all r), we will denote the underlying algebra as U q (g). This is a slight variation, obtained by considering 1 2 P instead of P , of the simply-connected version of the ordinary quantized universal enveloping algebra of g [4, Remark 9.1.3].
Remark that the algebras U q (g; ε, η) with r ε r η r = 0 are mutually isomorphic as unital algebras. Indeed, as A is invertible, we can choose b ∈ Char C (P ) such that b 4 αr = η r /ε r for all r ∈ Φ + . We can also choose a ∈ Char C (Q + ) such that a r is a square root of b 2 αr ε r = ηr b 2 αr . Then is a unital isomorphism. However, unless sgn(ε r η r ) = + for all r, in which case we can choose b ∈ Char R + (P ) and a ∈ Char R + (Q + ), this rescaling will not respect the * -structure.
We list some properties of the U q (g; ε, η). The proofs do not differ from those for the wellknown case U q (g), cf. [30] and [16,Section 4]. 1. Let U q (n ± ) be the unital subalgebra generated by the X ± r inside U q (g; ε, η). Then U q (n ± ) is the universal algebra generated by elements X ± r satisfying the relations (S) q , and in particular does not depend on ε or η.

2.
Similarly, let U q (b ± ) be the subalgebra generated by U q (n ± ) and U (h), where U (h) is the algebra generated by all K ω . Then U q (b ± ) is universal with respect to the relations (K) q , (T) q and (S) q .

(Triangular decomposition) The multiplication map gives an isomorphism
In particular, the U q (g; ε, η) are non-trivial. Note that the above proposition allows one to identify U q (g; ε, η) and U q (g) as vector spaces, Hence U q (g; ε, η) may also be viewed as U q (g) with a deformed product m ε,η . Indeed, arguing as in [21,Section 4], one can show that for certain cocycles ω ε on U q (g). Also the following proposition is immediate.
Proof . One immediately checks the coassociativity condition on the generators. One further can define uniquely a unital * -homomorphism ε : U q (g; ε, ε) → C, and unital anti-homomorphism Again, one verifies on generators that these maps satisfy the counit and antipode condition on generators, hence on all elements.
We will denote by ¡ the associated adjoint action of U q (g; η, η) on U q (g; ε, η), cf. Appendix A, Definition A.2.

Verma module theory
We keep the notation of the previous section.
Definition 2.1. For λ ∈ Char C (P ), we denote by C λ the one-dimensional left U q (b + )-module associated to the character We denote We denote by V ε,η λ the simple quotient of M ε,η λ by its maximal non-trivial submodule. Note that such a maximal submodule exists by the triangular decomposition (1.2), as then any non-trivial submodule is a sum of weight spaces with weights distinct from λ. We denote by v ε,η λ the highest weight vector 1 ⊗ 1 in either M ε,η λ or V ε,η λ . For λ ∈ Char R (P ), we can introduce on V ε,η λ a non-degenerate Hermitian form , such that xv, w = v, x * w for all x ∈ U q (g; ε, η) and v, w ∈ V ε,η λ . We will call a form satisfying this property invariant. Such a form is then unique up to a scalar. The construction of this form is the same as in the case of U q (g) [16,Section 5], [31, Section 2.1.5]. Namely, let U q (n ± ) + ⊆ U q (n ± ) be the kernel of the restriction of the counit on U q (b ± ), and consider the orthogonal decomposition Let τ denote the projection onto the first summand. Then one first observes that one has a Hermitian form on M ε,η λ by defining xv ε,η λ , yv ε,η λ = χ λ (τ (x * y)), x, y ∈ U q (g). This is clearly a well-defined invariant form. It necessarily descends to a non-degenerate Hermitian form on V ε,η λ . The goal is to find necessary and sufficient conditions for this Hermitian form to be positivedefinite, in which case the module is called unitarizable. This is in general a hard problem. In the following, we present some partial results, restricting to the case η = +. We always assume that λ is real-valued, unless otherwise mentioned, and that M ε,η λ and V ε,η λ have been equipped with the above canonical Hermitian form.
We first consider the case ε = η = +, for which the following result is well-known.
Lemma 2.2. Let λ ∈ Char R + (P ). Then the following are equivalent.  (2)). By a simple computation using the commutation between the X ± r (cf. [9]), we have that Hence, by unitarity and the fact that 0 < q r < 1, we find that k l=1 q l−1 r λ −2 αr − q −l+1 r λ 2 αr ≥ 0 for all k ≥ 0. This is only possible if eventually one of the factors becomes zero, i.e. when λ 4 αr ∈ q 2N r .
By a limit argument, we now extend this result to the case ε ∈ Char {0,1} (Q + ).
Proof . The 'only if' part of the proposition is obvious, since the K αr , X ± r with ε r = 1 generate a quantized enveloping algebra U q (k ss ) in its compact (non-simply connected) form.
To show the opposite direction, consider first general ε, η ∈ Char λ in the canonical way by means of x → xv ε,η λ , we may interpret the Hermitian forms on the M ε,η λ as a family of Hermitian forms ·, · ε,η λ on U q (n − ). It is easily seen that these vary continuously with ε, η and λ on each U q (n − )(α). Assume now that η = + and ε r > 0 for all r. In this case U q (g; ε, +) is isomorphic to U q (g) as a * -algebra by a rescaling of the generators by positive numbers, cf. (1.1). By Lemma 2.2, ·, · +,+ λ is positive semi-definite if and only if V +,+ λ is finite-dimensional, and the the latter happens if and only if λ 4 αr ∈ q 2N r for all r ∈ I. Hence we get by the above rescaling that ·, · ε,+ λ is positive semi-definite if and only if λ 4 αr ∈ q 2N r ε −1 r for all r. Fix now a subset S of the simple positive roots, and put ε r = 1 for r ∈ S. Assume that λ 4 αr ∈ q 2N r when r ∈ S. For r / ∈ S and m r ∈ N, define ε r = q 2mr r λ −4 αr . From the above, we obtain that ·, · ε,+ λ is positive semi-definite. Taking the limit m r → ∞ for r / ∈ S, we deduce that ·, · ε,+ λ is positive semi-definite for ε r = δ r∈S .
The above V ε,+ λ with pre-Hilbert space structure can also be presented more concretely as generalized Verma modules (from which it will be clear that they are not finite dimensional when ε r = 0 for some r). We will need some preparations, obtained from modifying arguments in [16]. Note that to pass from the conventions in [16] to ours, the q in [16] has to be replaced by q 1/2 , and the coproduct in [16] is also opposite to ours. However, as [16] gives preference to the left adjoint action, while we work with the right adjoint action, most of our formulas will in fact match.
We start with recalling a basic fact.
Proof . One checks that the arguments of [16, Lemma 6.1, Lemma 6.2, Proposition 6.3, Proposition 6.5] are still valid in our setting.
Let ε ∈ Char R (Q + ). Recall that the map τ denoted the projection onto the first summand in (2.1). Let τ Z,ε be the restriction of τ to the center Z (U q (g; ε, +)) of U q (g; ε, +). This is an ε-modified Harish-Chandra map. The usual reasoning shows that this is a homomorphism into U (h).
Lemma 2.5. The map τ Z,ε is a bijection between Z (U q (g; ε, +)) and the linear span of the set w∈W ε ω−wω q (−2wω,ρ) K −4wω ω ∈ P + , where ρ = r ω r is the sum of the fundamental weights and W denotes the Weyl group of ∆ with its natural action on h * .
Proof . Note first that ω − wω is inside Q + for all ω ∈ P + and w ∈ W , so that ε ω−wω is well-defined.
Arguing as in [16,Section 8], we can assign to any ω ∈ P + an element z ω in Z (U q (g; ε, +)), uniquely determined up to a non-zero scalar, such that where U q (g) + is the kernel of the counit and where ¡ denotes the adjoint action. More concretely, as K −4 ω ¡ U q (g) is a finite-dimensional right U q (g)-module, it is semi-simple, and we have a projection E of K −4 ω ¡ U q (g) onto the space of its invariant elements. We can then take z ω = E K −4 ω , and z ω ∈ Z (U q (g; ε, +)) by Lemma A.3. Suppose now first that ε r = 0 for all r, and choose b ∈ Char C (P ) such that b 4 This is a unital ¡-equivariant algebra isomorphism, and τ Z, Hence by ¡equivariance, we find from the computations in [16,Section 8] that, for a non-zero ε-independent scalar c ω , where (V ω ) ν denotes the weight space at q 1 2 ν (i.e. the space of vectors on which the K ω act as q 1 2 (ν,ω) ) of the finite-dimensional U q (g)-module V ω with highest weight q 1 2 ω . But clearly we then only have to sum over those ν with ω − ν ∈ Q + , so that b −4 ω−wν = ε ω−wν , and we can write Recall now that U q (g; ε, η) can be identified with U q (g) as a vector space by a map i ε,η . Let us denote by ¡ ε,η the image of ¡ under this map. Then for x, y ∈ U q (g) fixed, it is easily seen from the triangular decomposition that the x ¡ ε,η y live in a fixed finite-dimensional subspace of U q (g) as the ε, η vary, and the resulting map (ε, η) → x¡ ε,η y is then continuous. Furthermore, if V is a finite-dimensional right U q (g)-module with space of fixed elements V triv , we can find p ∈ U q (g) such that for any v ∈ V , the element vp is the projection of v onto V triv . It follows from the previous paragraph and the above remarks that when ε r = 0 for any r, we have i ε,+ (z ω ) = K −4 ω ¡ ε,+ p ω for some fixed p ω ∈ U q (g). By continuity, it then follows that (2.2) in fact holds for arbitrary ε ∈ Char R (Q + ).
The conclusion of the argument now follows as in [16,Theorem 8.6].
Let now S ⊆ Φ + , and let ε extend the characteristic function of S. Let U q (t S ) be the Hopf * -subalgebra of U q (g; ε, +) generated by the K ± ωr with 1 ≤ r ≤ l and X ± r with r ∈ S. Let U q (q + S ) be the Hopf subalgebra of U q (g; ε, +) generated by U q (t S ) and all X + r with r ∈ I. It is easy to see that U q (q + S ) can be isomorphically imbedded into U q (g). Let V be a finite-dimensional highest weight representation of U q (t S ) associated to a character in Char R + (P ). Then we can extend this to a representation of U q (q + S ) on V [31, Section 2.3.1], and hence we can form The following proposition complements Proposition 2.3.
Proposition 2.6. Let S ⊆ Φ + , and let ε restrict to the characteristic function of S. Let V be an irreducible highest weight representation of U q (t S ) associated to a character λ ∈ Char R + (P ). Then the U q (g; ε, +)-representation Ind ε (V ) is irreducible.
Proof . By its universal property, Ind ε (V ) can be identified with a quotient of M ε,+ λ . We want to show that this quotient coincides with V ε,+ λ . Suppose that v λ is a highest weight vector inside M ε,+ λ at weight λ different from λ. By Lemma 2.5, it follows that for all ω ∈ P + . Now if w = s α i 1 · · · s α ip in reduced form, with s α the reflection across the root α, we have where each term is positive. It follows that we have for all strictly dominant ω, where W S is the Coxeter group generated by reflections around simple roots α s with s ∈ S. Taking Hence λ ωr = λ ωr for all r / ∈ S. We deduce that v λ ∈ U q (t S )v λ , and so the image of v λ in Ind ε (V ) is zero. This implies that Ind ε (V ) = V ε,+ λ .
The case ε ∈ Char {−1,0,1} (Q + ) is not so easy to treat in general. In the following, we will restrict ourselves to the case where we have one ε t < 0 while ε r ≥ 0 for r = t. This will in particular comprise the 'symmetric Hermitian' case.
Proof . By a same kind of limiting argument as in Proposition 2.3, the general case can be deduced from the case with ε r = 1 for r = t.
Suppose then that ε t < 0 and ε r = 1 for r = t. Write S = Φ + \ {t}. We have the algebra automorphism φ : U q (g; ε, +) → U q (g) appearing in (1.1). By means of this isomorphism, we obtain a natural isomorphism M ε,+ λ ∼ = M +,+ γ , where γ ∈ Char C (P ) is such that γ 4 αr = ε r λ 4 αr for all r. In particular, γ 2 αt ∈ C \ R. It follows from [18,Proposition 5.13] that Ind ε (V λ ) is irreducible, where V λ denotes the irreducible representation of U q (q + S ) at highest weight λ. Hence the signatures of the Hermitian inner products on the Ind ε (V λ ) are constant as ε t < 0 varies. Indeed, these spaces can be identified canonically with a fixed quotient of U q (n − ), see [31,Proposition 2.81], and then the Hermitian inner products clearly form a continuous family as ε varies.
From Proposition 2.6, we know that the Hermitian inner product on Ind ε (V λ ) for ε t = 0 is positive definite. It follows that the Hermitian inner product on a weight space of Ind ε (V λ ) is positive for ε t < 0 small. As the signature is constant, this holds for all ε t < 0.

Quantized homogeneous spaces
Definition 3.1. For ε, η ∈ Char R (Q + ), we denote by U q (g; ε, η) fin the space of locally finite vectors in U q (g; ε, η) with respect to the right adjoint action by U q (g; η, η) (cf. Definition A.2), It is easily seen that the space U q (g; ε, η) fin is a * -subalgebra of U q (g; ε, η) (cf. [16, Corollary 2.3]), and in the following it will always be treated as a right U q (g)-module by ¡.
A similar definition of fin U q (g; ε, η) can be made with respect to the left adjoint action of U q (g; ε, ε), and the two resulting algebras U q (g; ε, η) fin and fin U q (g; ε, η) should in some sense be seen as dual to each other. For example, the U q (g; ε, +) fin will lead to compact quantum homogeneous spaces, while the fin U q (g; ε, +) should lead to non-compact quantum homogeneous spaces such as quantum bounded symmetric domains [31]. However, in this paper we will restrict ourselves to the compact case.
The U q (g; ε, η) fin are sufficiently large, as the next proposition shows, extending Lemma 2.4. Proposition 3.2. As a right U q (g)-module, U q (g; ε, +) fin is generated by the K −4 ω with ω ∈ P + . The algebra generated by U q (g; ε, +) fin and the K 4 ωr equals the subalgebra of U q (g; ε, +) generated by the K ±4 ωr and the K αr X ± r .
Proof . Again, the proof of [16, Theorem 6.4] can be directly modified.
Note that for ε r = 0 for all r, the above proposition follows more straightforwardly from [16] by a rescaling argument.
As the K −4 ω with ω ∈ P + generate U q (g; ε, +) fin as a module, ψ ε,η will be * -preserving if and only if b 4 αr ∈ R for all r. This can be realized if we can find c r ∈ {−1, 1} such that s c asr s = sgn(η r /ε r ), which is equivalent with the condition appearing in the statement of the lemma.
Remark 3.5. The space B fin λ (g; ε, η) is not defined as the space B λ (g; ε, η) fin of locally finite ¡-elements in B λ (g; ε, η), although conceivably they are the same in many cases. In the case q = 1, the equality of these two algebras goes by the name of the Kostant problem, cf. [19,Remark 3]. Notation 3.6. We will use the following notation for particular elements in the B λ (g; ε, +): The following commutation relations will be needed later on.
Lemma 3.7. The elements W r and T r commute with X r , Y r , Z r , T r and W r . Moreover, We further have that T r and W r are invariant under ¡X ± r and ¡K ω , while Finally, all elements X r , Y r , Z r , T r , W r are inside B fin λ (g; ε, +).
Remark 3.9. An alternative proof consists in applying Schur's lemma to the simple module V ε,+ λ . Indeed, x ∈ B fin λ (g; ε, +) is ¡-invariant if and only if it commutes with all π ε,+ λ (y) for y ∈ U q (g; ε, +). As V ε,+ λ is simple, Schur's lemma implies that the algebra of ¡-invariant elements in B fin λ (g; ε, +) forms a field of countable dimension over C, hence coincides with C. (I would like the referee for pointing out this approach). Proposition 3.10. Let (V, π) be a * -representation of B fin λ (g; ε, +) on a pre-Hilbert space. Then π is bounded.
The proof is based on an argument which is well-known in the setting of compact quantum groups.
Proof . As B fin λ (g; ε, +) consists of locally finite elements, any b ∈ B fin λ (g; ε, +) can be written as a finite linear combination of elements b i ∈ B fin λ (g; ε, +) for which there exists a finite-dimensional * -representation π of U q (g) on a Hilbert space such that b i ¡ h = j π ij (h)b j for all h ∈ U q (g), the π ij being the matrix components with respect to some orthogonal basis. An easy computation shows that i b * i b i is an invariant element, hence a scalar by Proposition 3.8. Hence there exists C ∈ R + such that for any ξ ∈ V and any i, we have π(b i )ξ ≤ C ξ . We deduce that the element π(b) is bounded.
Definition 3.11. A B fin λ (g; ε, +)-module V is called a highest weight module if there exists a cyclic vector v ∈ V which is annihilated by all X r and which is an eigenvector for all Z r with non-zero eigenvalue. A pre-Hilbert space structure on V is called invariant if xξ, η = ξ, x * η for all ξ, η ∈ V and x ∈ B fin λ (g; ε, +).
We aim to show that the B fin λ (g; ε, +) have only a finite number of non-equivalent irreducible highest weight modules. Of course, each B fin λ (g; ε, +) admits at least the highest weight module V ε,+ λ . Also note that, by an easy argument, each highest weight module decomposes into a direct sum of joint weight spaces for the Z r . Proposition 3.12. Each B fin λ (g; ε, +) admits only a finite number of non-equivalent irreducible highest weight modules.
Proof . As the statement does not depend on the * -structure, we may by rescaling restrict to the case that ε r ∈ {0, 1} for all r upon allowing λ ∈ Char C (P ).
By Proposition 3.2 and the fact that any highest weight module is semi-simple for the torus part, it is easily argued that any irreducible highest weight module of B fin λ (g; ε, +) is obtained by restriction of a U q (g; ε, +)-module V ε,+ λ for some λ ∈ Char C (P ). As the center of U q (g; ε, +) acts by the same character on V ε,+ λ and V ε,+ λ , we find by Lemma 2.5 that the expression w∈W ε ω−wω q −2(wω,ρ) λ −4wω remains the same upon replacing λ by λ , for each ω ∈ P + . Writing S for the set of r with ε r = 0, it follows as in the proof of Lemma 2.5 that for all ω ∈ P ++ , the strictly dominant weights. As (invertible) characters on a commutative semi-group are linearly independent, and as P ++ − P ++ = P , it follows that the functions ω → q −2(ω,ρ) λ −4ω and ω → q −2(ω,ρ) λ −4ω on P lie in the same W S -orbit. As the highest weight vector in an irreducible highest weight module is uniquely determined up to a scalar, and as the equivalence classes of such highest weight modules are then determined by the associated eigenvalue of the Z r = π ε,+ λ (K −4 ωr ), this is sufficient to prove the proposition.
Remark that the above proof also gives the upper bound |W S | for the number of inequivalent highest weight representations, but of course this estimate is not sharp if one only considers unitarizable representations. Proposition 3.13. Let π be a * -representation of B fin λ (g; ε, +) on a Hilbert space H . If 0 is not in the point-spectrum of any of the Z r , then H is a (possibly infinite) direct sum of completions (H k , π k ) of unitarizable highest weight modules of B fin λ (g; ε, +).
Proof . By a direct integral decomposition, and using Proposition 3.12, it is sufficient to show that any such irreducible * -representation π of B fin λ (g; ε, +) on a Hilbert space H is the completion of a highest weight module with invariant pre-Hilbert space structure. We then argue as in [27,Section 3]. Write χ X for the characteristic function of a set. By assumption, there exists t ∈ R l with t r = 0 for all r and P t = χ r [qrtr,tr] (π(Z 1 ), . . . , π(Z l )) non-zero. Suppose now that r is such that π(X r )P t = 0. From the commutation relations between the X r and the Z s , we deduce that P (t 1 ,...,q −2 r tr,...,t l ) = 0. As the π(Z r ) are bounded, this process must necessarily stop. Hence we may choose t such that P t = 0 but π(X r )P t = 0 for all r.
Let V be the union of the images of the spectral projections of (Z 1 , . . . , Z l ) corresponding to the r R \ (− 1 n , 1 n ) with n ∈ N. As B fin λ (g; ε, +) is spanned by elements which skew-commute with the Z r , it follows that V is a B fin λ (g; ε, +)-module on which the π(Z r ) are invertible linear maps. This entails that the restriction of π to V can be extended to a representation π of B fin (g; ε, +) ext , the sub * -algebra of B(g; ε, +) generated by X r , Y r and the Z ±1 r (which contains B fin λ (g; ε, +) by Proposition 3.2). Note that this * -algebra admits a triangular decomposition (in the obvious way with respect to the above generators).
Pick now a non-zero ξ ∈ P t H . Suppose that ξ were not in the pure point spectrum of some π(Z r ). Then we can find q r t r < a < t r such that χ [qrtr,a] (π(Z r ))ξ = 0 = χ (a,tr] (π(Z r ))ξ.
s t s , q 2ks s t s ] = ∅ for all k s ∈ N with at least one k s > 0. From the commutation relations between the Y s and Z s , and the fact that π(X s )ξ = 0 for all s, we deduce that χ [qrtr,a] (π(Z r ))ξ is orthogonal to the B fin λ (g; ε, +) ext -module spanned by χ (a,tr] (π(Z r ))ξ. As π is irreducible, this would entail χ [qrtr,a] (π(Z r ))ξ = 0. Having arrived at a contradiction, we conclude that ξ is a joint eigenvector of all π(Z r ).
As ξ is annihilated by all π(X r ) and is a joint eigenvector of all π(Z r ), the module generated by it is a highest weight module. As π was irreducible, this module must necessarily be dense in H , and the proposition is proven.
We now want to consider analytic versions of the B fin λ (g; ε, +).
Definition 3.14. Let B be a unital * -algebra. We say that B admits a universal C * -envelope if there exists a non-trivial unital C * -algebra C together with a unital * -homomorphism π u : B → C of unital * -algebras such that any * -homomorphism B → D with D a unital C * -algebra factors through C.
Of course, the above C * -algebra C is then uniquely determined up to isomorphism.
Definition 3.15. We define Pol(G q + ) to be the Hopf * -algebra inside the dual of U q (g) which is spanned by the matrix coefficients of finite-dimensional highest weight representations of U q (g) associated to positive characters. We define as the comodule * -algebra structure dual to the module * -algebra structure ¡ by U q (g).
Note that the latter definition makes sense, since B fin λ (g; ε, +) is integrable as a right U q (g)module.
It is known [26] that Pol(G q + ) admits a universal C * -algebraic envelope C(G q + ), which becomes a compact quantum group in the sense of [32]. We will denote by ϕ G q + the invariant state on C(G q + ), which is faithful by co-amenability of G q + .
Proof . The universal C * -algebraic envelope of B fin λ (g; ε, +) exists for precisely the same reason as in Proposition 3.10, since for any element b ∈ B fin λ (g; ε, +) there exists a universal constant C b such that π(b) ≤ C for all * -representations π of B fin λ (g; ε, +) by bounded operators on a Hilbert space. As B fin λ (g; ε, +) admits at least one * -representation, we have C λ (g; ε, +) = 0.
The proof will make use of the following standard lemma. Lemma 3.21. Let A be a unital C * -algebra with faithful state ϕ. Let M be the von Neumann algebra closure of A in its GNS-representation with respect to ϕ. Let π be a representation of A on a Hilbert space H such that there exists a faithful state ω ∈ B(H ) * with ω • π = ϕ. Then π extends to a normal faithful * -representation of M .
To prove the remaining part of the theorem, we first make some preparations. Recall that the dual of U q (sl(2, C)) can be identified with Pol(SU q (2)), Woronowicz's twisted quantum SU (2)-group [33]. It is well-known that its von Neumann algebraic envelope L ∞ (SU q (2)) is isomorphic to B(l 2 (N)) ⊗ L (Z), and as such admits a faithful representation on the Hilbert space H + = l 2 (N) ⊗ l 2 (Z) (cf. [26,29]).
More generally, write U qr (su (2)) for the sub-Hopf- * -algebra generated by the X ± r and K ±1 αr inside U q (g). By duality, one obtains a surjective * -homomorphism γ r : Pol(G q + ) → Pol(SU qr (2)). This induces a * -representation of Pol(G q + ) on H + , which we will denote by the same symbol γ r .
Suppose now that t = (r 1 , . . . , r n ) is an ordered n-tuple of elements in I. Then we obtain a * -representation of Pol(G q + ) on H ⊗n + by means of the * -homomorphism denotes the n-fold coproduct. Let now t 0 = (r 1 , . . . , r N ) be such that w 0 = s r 1 · · · s r N is a reduced expression for the longest element in the Weyl group of g. By [29], we know that ϕ G q + can be realized as ω • γ t 0 for some faithful normal state ω ∈ B H ⊗N + * . By Lemma 3.21, this implies that γ t 0 can be extended to a faithful normal * -representation of L ∞ (G q + ). Let us turn now to the proof of the theorem. The main step is to prove that the pointspectrum of θ ε,+ λ,reg l r=1 Z r does not contain zero. Indeed, if this is the case, then we can use Proposition 3.13 to conclude that θ ε,+ λ,reg decomposes into a direct integral of highest weight modules for B fin λ (g; ε, +). It then follows that W λ (g; ε, +) is simply ⊕ m k=1 B(H k ) with H k the Hilbert space completion of the highest weight modules which appear in θ ε,+ λ,reg .
Finally, to show that (θ ε,+ λ ⊗γ t 0 ) α ε,+ λ l r=1 Z r does not contain zero in its pointspectrum, we can reason by induction, using the following lemma.
Lemma 3.22. Let (V, π) be an irreducible highest weight module for B fin λ (g; ε, +) with an invariant pre-Hilbert space structure, and let H be the completion of V . Fix r ∈ I, and put π r = (π ⊗ γ r )α ε,+ λ . Then π r l s=1 Z s does not contain 0 in its point spectrum.
Proof . It is easy to see that π r (Z s ) = 1 ⊗ π(Z s ) for r = s, so that none of these operators have zero in their point-spectrum. Let now A r be the sub- * -algebra of B fin λ (g; ε, +) generated by Z r , T r , X r , Y r , W r , see Lemma 3.7. By that lemma, A r is stable under the right action by U qr (su(2)), and W r and T r are invariants in the center of A r . By invariance, π r (W r ) = 1 ⊗ π(W r ) and π r (T r ) = 1 ⊗ π(T r ) are bounded self-adjoint operators. Hence, to investigate the spectrum of π r (Z r ), we may by disintegration treat the above operators as scalars, say w r and t r . Denote the resulting quotient of A r by A r (w r , t r ).
From Lemma 3.7, we find commutation relations between the generators X r , Y r and Z r of A r (w r , t r ), as well as the resulting action of U qr (su(2)). It follows that A r (w r , t r ) is an equivariant quotient of a generalized Podleś sphere S 2 qr,τ for SU qr (2) [27], for some τ depending on t r and w r . Moreover, as ϕ SUq r (2) can be realized on the Hilbert space H + , the von Neumann algebraic envelope of A r (w r , t r ) will be isomorphic to the von Neumann algebraic envelope of S 2 qr,τ , which is equal to M n (C), B(l 2 (N)) or B(l 2 (N)) ⊕ B(l 2 (N)), depending on whether ε r w r is positive, zero or negative (cf. [27]). In any case, the corresponding image of Z r will not contain 0 in its point-spectrum. 4 More on the ε ∈ Char {0,1} (Q + )-case The case of the non-standard Podleś spheres already shows that W λ (g; ε, +) is in general not a factor. However, for ε ∈ Char {0,1} (Q + ) and λ ∈ Char R + (P ) such that W λ (g; ε, +) is welldefined, we show that W λ (g; ε, +) does become a type I-factor, and we can then also say something more about the invariant integral ϕ ε,+ λ on W λ (g; ε, +).
Proof . From the proof of Theorem 3.20 and Lemma 3.22, and from the commutation relations in Lemma 3.7, we get that the A r (w r , t r ) appearing in the proof of Lemma 3.22 can only be matrix algebras or standard Podleś spheres. As the Z r in the von Neumann algebraic completion of these algebras are always positive operators, it follows by induction that the components appearing in W λ (g; ε, +) arise from restrictions of highest weight modules V ε,+ λ of U q (g; ε, +) with λ ∈ Char R + (P ). Our aim is to show that necessarily λ = λ.
Let S be the set of r with ε r = 1. Suppose that λ ∈ Char R + (P ) is such that the representation of U q (g; ε, +) on V ε,+ λ factors over B fin λ (g; ε, +). Suppose that V ε,+ λ then admits an invariant pre-Hilbert space structure as a B fin λ (g; ε, +)-module, hence as a U q (g; ε, +)-module by Proposition 3.2. From the proof of Proposition 3.12, we deduce that there exists w ∈ W S such that q (−2wω,ρ) λ −4wω = q (−2ω,ρ) λ −4ω for all ω ∈ P . In particular, λ ωr = λ ωr for r / ∈ S. On the other hand, let U q (k ss ) be the subalgebra of U q (g; ε, +) generated by the X ± r and K ±1 αr with r ∈ S. Let V ε,+ λ be the U q (k ss )-module spanned by v ε,+ λ , and similarly for V ε,+ λ . Then by Proposition 2.6, these are irreducible highest weight modules for U q (k ss ) associated to the restrictions of λ and λ to the root lattice Q S of k ss . But as V ε,+ λ admits an invariant pre-Hilbert space structure (and k ss is compact), it is necessarily finite-dimensional. However, as the restriction of λ lies in the W S -orbit of λ (for the so-called 'dot'-action), it is well-known that this can happen only if the restrictions of λ and λ to Q S coincide. Combined with the observation at the end of the previous paragraph, this forces λ = λ inside Char R + (P ).

A Cogroupoids
In this section, we recall the notion of cogroupoid due to J. Bichon [1,2] (cf. also the notion of face algebra [13]).
Definition A.1. Let I be an index set, and let {H ij | i, j ∈ I} be a collection of * -algebras. Suppose that for each triple of indices i, j, k ∈ I, we are given a unital * -homomorphism We call {H ij }, {∆ k ij } a connected cogroupoid over the index set I if the following conditions are satisfied: • (Connectedness) None of the H ij are the zero algebra.
• (Coassociativity) For each quadruple i, j, k, l of indices, we have • (Counits) There exist unital * -homomorphisms i : H ii → C such that, for all indices i, j, • (Antipodes) There exist anti-homomorphisms S ij : H ij → H ji such that, for all indices i, j and all h ∈ H ii , we have As for Hopf * -algebras, it is easy to show that the S ij are unique, and that S ji (S ij (h) * ) * = h for each h ∈ H ij . Note that each H ii , ∆ i ii defines a Hopf * -algebra. Definition A.2. Let H ij , ∆ k ij be a cogroupoid. The right adjoint action (or Miyashita-Ulbrich action) ¡ of H jj on H ij is given by x ¡ h = S ji (h (1)ji )xh (2)ij .
One easily proves that ¡ defines a right H jj -module * -algebra structure on H ij . The compatibility with the * -structure means that (x ¡ h) * = x * ¡ S jj (h) * .
Lemma A.3. The space of ¡-invariant elements in H ij coincides with the center of H ij .
Suppose now that B is a unital * -algebra, and that for some i, j we have a unital * -homomorphism π : H ij → B. As the right action of H jj on H ij is inner, it descends to a right action on B, b ¡ h = π(S ji (h (1)ji ))bπ(h (2)ij ).
Note that the central elements in B are invariant for the action.

B Continuous one-parameter families of Lie algebras
We introduce the real Lie algebras whose quantizations we studied in Section 1. We refer to standard works as [14,15,22] for the basic background on Lie algebras and Lie groups.
We keep the notation as in Section 1. We further write {h r } ⊆ h for the basis dual to {ω r }, and write h R for its real span, which we may identify with the real dual of h * R . We write g c ⊆ g for the compact real form of g, and † for the corresponding anti-linear anti-automorphism such that x † = −x for x ∈ g c .
Finally, for each α ∈ ∆ + , we choose root vectors X ± α ∈ n ± such that (X + α ) † = X − α for all α ∈ ∆ + and [X + r , X − r ] = h r for all α r ∈ Φ + . Fix ε ∈ Char R (Q + ). We can define on g the linear maps S ± ε : g → g : Definition B.1. We define g ε to be the complex vector space Proposition B.2. The vector space g ε is a Lie * -subalgebra of the direct sum Lie algebra g ⊕ g equipped with the involution (w, z) * = (z † , w † ). Moreover, dim C (g ε ) = dim C (g).
Proof . From the definition, we see that g ε is the linear space generated by elements of the form (ε α X + α , X + α ), (X − α , ε α X − α ) and (h r , h r ). Accordingly, g ε has dimension dim C (g) and inherits the * -operation from g ⊕ g. Since ε α+β = ε α ε β whenever α, β and α + β are positive roots, we find that g ε is closed under the bracket operation.
In the following, we consider g ε with its * -operation inherited from g ⊕ g.
Remark B.3. By rescaling the X ± α , we see that g ε ∼ = g η as Lie * -algebras whenever ε r = λ r η r for certain λ r > 0. Hence we may in principle always assume that ε ∈ {−1, 0, 1} l . It is however sometimes convenient to keep the continuous deformation aspect into the game. In the physics literature, this type of deformation goes by the name of 'the contraction method'. (In low dimensions, it can easily be visualized, cf. [12,Chapter 13].) Definition B.4. We define g R ε = {z ∈ g ε | z * = −z}.
Hence g R ε is a real Lie algebra with g ε as its complexification. We can also realize g R ε more conveniently inside g as follows.
Proposition B.5. Write X (ε) α = X + α − ε α X − α , Y (ε) α = i X + α + ε α X − α as elements in g. Consider the R-linear span of the X α and ih r . Then this space is closed under the Lie bracket of g, and forms a real Lie algebra isomorphic to g R ε .