Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

Let g be a compact simple Lie algebra. We modify the quantized enveloping ∗-algebra associated to 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. Restricti...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2013
1. Verfasser:	de Commer, K.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2013
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/149373
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure / K. de Commer // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 33 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-149373
record_format	dspace
spelling	irk-123456789-1493732019-02-22T01:23:38Z Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure de Commer, K. Let g be a compact simple Lie algebra. We modify the quantized enveloping ∗-algebra associated to 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. 2013 Article Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure / K. de Commer // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 33 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 20G42; 46L65 DOI: http://dx.doi.org/10.3842/SIGMA.2013.081 http://dspace.nbuv.gov.ua/handle/123456789/149373 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let g be a compact simple Lie algebra. We modify the quantized enveloping ∗-algebra associated to 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.
format	Article
author	de Commer, K.
spellingShingle	de Commer, K. Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure Symmetry, Integrability and Geometry: Methods and Applications
author_facet	de Commer, K.
author_sort	de Commer, K.
title	Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
title_short	Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
title_full	Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
title_fullStr	Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
title_full_unstemmed	Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
title_sort	representation theory of quantized enveloping algebras with interpolating real structure
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149373
citation_txt	Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure / K. de Commer // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 33 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT decommerk representationtheoryofquantizedenvelopingalgebraswithinterpolatingrealstructure
first_indexed	2025-07-12T21:58:47Z
last_indexed	2025-07-12T21:58:47Z
_version_	1837480053644984320
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 081, 20 pages Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure? Kenny DE COMMER Department of Mathematics, University of Cergy-Pontoise, UMR CNRS 8088, F-95000 Cergy-Pontoise, France E-mail: Kenny.De-Commer@u-cergy.fr URL: http://kdecommer.u-cergy.fr Received August 18, 2013, in final form December 18, 2013; Published online December 24, 2013 http://dx.doi.org/10.3842/SIGMA.2013.081 Abstract. Let g be a compact simple Lie algebra. We modify the quantized enveloping ∗-algebra associated to 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. Key words: compact quantum homogeneous spaces; quantized universal enveloping algebras; Hopf–Galois theory; Verma modules 2010 Mathematics Subject Classification: 17B37; 20G42; 46L65 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 mailto:Kenny.De-Commer@u-cergy.fr http://kdecommer.u-cergy.fr http://dx.doi.org/10.3842/SIGMA.2013.081 http://www.emis.de/journals/SIGMA/Rieffel.html 2 K. De Commer 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 homoge- neous 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+ = {αr \| 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 = (ars)r,s∈I = ((α∨r , αs))r,s∈I is the Cartan matrix of g, where α∨ = 2 (α,α)α for α ∈ ∆. Let hR ⊆ h be the real linear span of the coroots hα, where β(hα) = (β, α∨) for α, β ∈ ∆. We further use the following notation. We write {ωr \| r ∈ I} for the fundamental weights in h∗R, so (ωr, α ∨ s ) = δrs. The Z−lattice spanned by {ωr} is denoted P ⊆ h∗R, and P+ denotes elements expressed as positive linear combinations of this basis. Similarly, the root lattice spanned by the αr is denoted Q, and its positive span by Q+. We write CharK(F ) for the monoid of monoid homomorphisms from a commutative (additive) monoid (F,+) to a commutative (multiplicative) monoid (K, ·). For ε ∈ CharK(Q+) we will abbreviate εαr = εr. The unit element of CharK(Q+) will be denoted +, while the element ε such that εr = 0 for all r will be denoted 0. Representations of Interpolating QUE Algebras 3 We use the following notation for q-numbers, where 0 < q < 1 is fixed for the rest of the paper: • for r ∈ Φ+, qr = q (αr,αr) 2 , • for n ≥ 0, [n]r = qnr−q −n r qr−q−1 r , • for n ≥ 0, [n]r! = n∏ k=1 [k]r, • for m ≥ n ≥ 0, [ m n ] r = [m]r! [n]r![m−n]r! , • for α ∈ h∗, we define qα ∈ CharC(P ) by qα(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 S−1(h) = S(h∗)∗ for all h ∈ H. Definition 1.1. For ε, η ∈ CharR(Q+), we define Uq(g; ε, η) as the universal unital ∗-algebra generated by couples of elements X±r , r ∈ I, as well as elements Kω, ω ∈ P , such that for all r, s ∈ I and ω, χ ∈ P , we have Kω self-adjoint, (X+ r )∗ = X−r and (K)q Kω is invertible and KχK −1 ω = Kχ−ω, (T)q KωX ± r K −1 ω = q ±(ω,αr) 2 X±r , (S)q 1−ars∑ k=0 (−1)k [ 1− ars k ] r (X±r )kX±s (X±r )1−ars−k = 0 for r 6= s, (C)qε,η [X+ r , X − s ] = δrs εrK2 αr −ηrK−2 αr qr−q−1 r . When ε = η = + (i.e. εr = ηr = 1 for all r), we will denote the underlying algebra as Uq(g). This is a slight variation, obtained by considering 1 2P instead of P , of the simply-connected version of the ordinary quantized universal enveloping algebra of g [4, Remark 9.1.3]. The Uq(g; ε, ε) are quantizations of the ∗-algebras U(gε), introduced in Appendix B. Note also that the algebras Uq(g; 0,+) are well-known to algebraists, see [20, Section 3] and [17]. Remark that the algebras Uq(g; ε, η) with ∏ r εrηr 6= 0 are mutually isomorphic as unital algebras. Indeed, as A is invertible, we can choose b ∈ CharC(P ) such that b4αr = ηr/εr for all r ∈ Φ+. We can also choose a ∈ CharC(Q+) such that ar is a square root of b2αrεr = ηr b2αr . Then φ : Uq(g; ε, η)→ Uq(g),  X+ r 7→ arX + r , X−r 7→ arX − r , Kω 7→ bωKω (1.1) is a unital isomorphism. However, unless sgn(εrηr) = + for all r, in which case we can choose b ∈ CharR+(P ) and a ∈ CharR+(Q+), this rescaling will not respect the ∗-structure. We list some properties of the Uq(g; ε, η). The proofs do not differ from those for the well- known case Uq(g), cf. [30] and [16, Section 4]. 4 K. De Commer Proposition 1.2. 1. Let Uq(n ±) be the unital subalgebra generated by the X±r inside Uq(g; ε, η). Then Uq(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 Uq(b ±) be the subalgebra generated by Uq(n ±) and U(h), where U(h) is the algebra generated by all Kω. Then Uq(b ±) is universal with respect to the relations (K)q, (T)q and (S)q. 3. (Triangular decomposition) The multiplication map gives an isomorphism Uq ( n+ ) ⊗ U(h)⊗ Uq(n−)→ Uq(g; ε, η). (1.2) In particular, the Uq(g; ε, η) are non-trivial. Note that the above proposition allows one to identify Uq(g; ε, η) and Uq(g) as vector spaces, iε,η : Uq(g; ε, η)→ Uq(g), x+x0x− 7→ x+x0x−, x± ∈ Uq ( n± ) , x0 ∈ U(h). Hence Uq(g; ε, η) may also be viewed as Uq(g) with a deformed product mε,η. Indeed, arguing as in [21, Section 4], one can show that mε,η(x⊗ y) = ωε(x(1), y(1))x(2)y(2)ωη(x(3), y(3)) for certain cocycles ωε on Uq(g). Also the following proposition is immediate. Proposition 1.3. For each ε, µ, η ∈ CharR(Q+), there exists a unique unital ∗-homomorphism ∆µ ε,η : Uq(g; ε, η)→ Uq(g; ε, µ)⊗ Uq(g;µ, η) such that ∆µ ε,η(Kω) = Kω ⊗Kω for all ω ∈ P and ∆µ ε,η ( X±r ) = X±r ⊗Kαr +K−1αr ⊗X ± r , ∀ r ∈ Φ+. Proof. From Proposition 1.2 and the case ε = η = +, we know that ∆µ ε,η respects the relations (K)q, (T)q and (S)q. The compatibility with the relations (C)qε,η is verified directly. � Lemma 1.4. The collection ( {Uq(g; ε, η)}, {∆µ ε,η} ) forms a connected cogroupoid over the index set CharR(Q+) ∼= Rl (cf. Appendix A). Proof. One immediately checks the coassociativity condition on the generators. One further can define uniquely a unital ∗-homomorphism εε : Uq(g; ε, ε)→ C, { X±r 7→ 0, Kω 7→ 1 and unital anti-homomorphism Sε,η : Uq(g; ε, η)→ Uq(g; η, ε), { X±r 7→ −q±1X±r , Kω 7→ K−1ω . 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 Uq(g; η, η) on Uq(g; ε, η), cf. Appendix A, Definition A.2. Representations of Interpolating QUE Algebras 5 2 Verma module theory We keep the notation of the previous section. Definition 2.1. For λ ∈ CharC(P ), we denote by Cλ the one-dimensional left Uq(b +)-module associated to the character χλ : Uq ( b+ ) → C, { X+ r 7→ 0, Kω 7→ λω. We denote M ε,η λ = Uq(g; ε, η) ⊗ Uq(b+) Cλ. 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 vε,ηλ the highest weight vector 1⊗ 1 in either M ε,η λ or V ε,η λ . For λ ∈ CharR(P ), we can introduce on V ε,η λ a non-degenerate Hermitian form 〈 , 〉 such that 〈xv,w〉 = 〈v, x∗w〉 for all x ∈ Uq(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 Uq(g) [16, Section 5], [31, Section 2.1.5]. Namely, let Uq(n ±)+ ⊆ Uq(n±) be the kernel of the restriction of the counit on Uq(b ±), and consider the orthogonal decomposition Uq(g; ε, η) = U(h)⊕ ( Uq ( n− ) + Uq(g; ε, η) + Uq(g; ε, η)Uq(n +)+ ) . (2.1) 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 〈xvε,ηλ , yvε,ηλ 〉 = χλ(τ(x∗y)), x, y ∈ Uq(g). This is clearly a well-defined invariant form. It necessarily descends to a non-degenerate Her- mitian form on V ε,η λ . The goal is to find necessary and sufficient conditions for this Hermitian form to be positive- definite, 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 ∈ CharR+(P ). Then the following are equivalent. 1. The module V +,+ λ is unitarizable. 2. For all r ∈ I, λ4αr ∈ q 2N r . 3. V +,+ λ is finite-dimensional. Proof. For the implications (2)⇒ (3)⇒ (1), see e.g. [4, Corollary 10.1.15, Proposition 10.1.21]. Assume now that V +,+ λ is unitarizable. Then V +,+ λr is a unitarizable representation of Uqr(su(2)). By a simple computation using the commutation between the X±r (cf. [9]), we have that 〈(X+ r )k(X−r )kv+,+λr , v+,+λr 〉 = k∏ l=1 ( qlr − q−lr )( q−l+1 r λ2αr − q l−1 r λ−2αr )( qr − q−1r )2 . 6 K. De Commer Hence, by unitarity and the fact that 0 < qr < 1, we find that k∏ l=1 ( ql−1r λ−2αr − q −l+1 r λ2αr ) ≥ 0 for all k ≥ 0. This is only possible if eventually one of the factors becomes zero, i.e. when λ4αr ∈ q 2N r . � By a limit argument, we now extend this result to the case ε ∈ Char{0,1}(Q +). Proposition 2.3. Suppose ε ∈ Char{0,1}(Q +), and let λ ∈ CharR+(P ). Then V ε,+ λ is unitari- zable if and only if λ4αr ∈ q 2N r for all r ∈ I with εr = 1. Proof. The ‘only if’ part of the proposition is obvious, since the Kαr , X ± r with εr = 1 generate a quantized enveloping algebra Ũq(kss) in its compact (non-simply connected) form. To show the opposite direction, consider first general ε, η ∈ CharR(Q+). For α ∈ Q+, denote by Uq(n −)(α) the finite-dimensional space of elements X ∈ Uq(n−) with KωXK −1 ω = q− 1 2 (ω,α)X. Identifying Uq(n −) with M ε,η λ in the canonical way by means of x 7→ xvε,ηλ , we may interpret the Hermitian forms on the M ε,η λ as a family of Hermitian forms 〈 ·, · 〉ε,ηλ on Uq(n −). It is easily seen that these vary continuously with ε, η and λ on each Uq(n −)(α). Assume now that η = + and εr > 0 for all r. In this case Uq(g; ε,+) is isomorphic to Uq(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 λ4αr ∈ 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 λ4αr ∈ q 2N r ε−1r for all r. Fix now a subset S of the simple positive roots, and put εr = 1 for r ∈ S. Assume that λ4αr ∈ q 2N r when r ∈ S. For r /∈ S and mr ∈ N, define εr = q2mrr λ−4αr . From the above, we obtain that 〈 ·, · 〉ε,+λ is positive semi-definite. Taking the limit mr → ∞ for r /∈ S, we deduce that 〈 ·, · 〉ε,+λ is positive semi-definite for εr = δr∈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 εr = 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 q1/2, 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. Lemma 2.4. Let ω ∈ P+, ε ∈ CharR(Q+), and consider K−4ω ∈ Uq(g; ε,+). Then K−4ω �Uq(g) is finite-dimensional, where � is the adjoint action (cf. Definition A.2). Proof. One checks that the arguments of [16, Lemma 6.1, Lemma 6.2, Proposition 6.3, Propo- sition 6.5] are still valid in our setting. � Let ε ∈ CharR(Q+). Recall that the map τ denoted the projection onto the first summand in (2.1). Let τZ,ε be the restriction of τ to the center Z (Uq(g; ε,+)) of Uq(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 τZ,ε is a bijection between Z (Uq(g; ε,+)) and the linear span of the set{ ∑ w∈W εω−wωq (−2wω,ρ)K−4wω ∣∣∣ω ∈ P+ } , where ρ = ∑ r ωr is the sum of the fundamental weights and W denotes the Weyl group of ∆ with its natural action on h∗. Representations of Interpolating QUE Algebras 7 Proof. Note first that ω − wω is inside Q+ for all ω ∈ P+ and w ∈ W , so that εω−wω is well-defined. Arguing as in [16, Section 8], we can assign to any ω ∈ P+ an element zω in Z (Uq(g; ε,+)), uniquely determined up to a non-zero scalar, such that K−4ω � Uq(g) = Czω + ( K−4ω � Uq(g)+ ) , where Uq(g)+ is the kernel of the counit and where � denotes the adjoint action. More con- cretely, as K−4ω � Uq(g) is a finite-dimensional right Uq(g)-module, it is semi-simple, and we have a projection E of K−4ω � Uq(g) onto the space of its invariant elements. We can then take zω = E ( K−4ω ) , and zω ∈ Z (Uq(g; ε,+)) by Lemma A.3. Suppose now first that εr 6= 0 for all r, and choose b ∈ CharC(P ) such that b4αr = ε−1r . Consider Ψε : Uq(g; ε,+)→ Uq(g), { X±r 7→ b−1αrX ± r , Kβ 7→ bβKβ, β ∈ P. This is a unital �-equivariant algebra isomorphism, and τZ,ε = Ψ−1ε ◦ τZ,+ ◦ Ψε. Hence by �- equivariance, we find from the computations in [16, Section 8] that, for a non-zero ε-independent scalar cω, τZ,ε(zω) = cω ∑ ν∈P+ dim((Vω)ν) (∑ w∈W b−4ω−wνq −2(wν,ρ)K−4wν ) , where (Vω)ν denotes the weight space at q 1 2 ν (i.e. the space of vectors on which the Kω act as q 1 2 (ν,ω)) of the finite-dimensional Uq(g)-module Vω 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−4ω−wν = εω−wν , and we can write τZ,ε(zω) = cω ∑ ν∈P+ ω−ν∈Q+ dim((Vω)ν) (∑ w∈W εω−wνq −2(wν,ρ)K−4wν ) . (2.2) Recall now that Uq(g; ε, η) can be identified with Uq(g) as a vector space by a map iε,η. Let us denote by �ε,η the image of � under this map. Then for x, y ∈ Uq(g) fixed, it is easily seen from the triangular decomposition that the x �ε,η y live in a fixed finite-dimensional subspace of Uq(g) as the ε, η vary, and the resulting map (ε, η) 7→ x�ε,ηy is then continuous. Furthermore, if V is a finite-dimensional right Uq(g)-module with space of fixed elements Vtriv, we can find p ∈ Uq(g) such that for any v ∈ V , the element vp is the projection of v onto Vtriv. It follows from the previous paragraph and the above remarks that when εr 6= 0 for any r, we have iε,+(zω) = K−4ω �ε,+ pω for some fixed pω ∈ Uq(g). By continuity, it then follows that (2.2) in fact holds for arbitrary ε ∈ CharR(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 Uq(tS) be the Hopf ∗-subalgebra of Uq(g; ε,+) generated by the K±ωr with 1 ≤ r ≤ l and X±r with r ∈ S. Let Uq(q + S ) be the Hopf subalgebra of Uq(g; ε,+) generated by Uq(tS) and all X+ r with r ∈ I. It is easy to see that Uq(q + S ) can be isomorphically imbedded into Uq(g). Let V be a finite-dimensional highest weight representation of Uq(tS) associated to a character in CharR+(P ). Then we can extend this to a representation of Uq(q + S ) on V [31, Section 2.3.1], and hence we can form Indε(V ) = Uq(g; ε,+) ⊗ Uq(q + S ) V. The following proposition complements Proposition 2.3. 8 K. De Commer 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 Uq(tS) associated to a character λ ∈ CharR+(P ). Then the Uq(g; ε,+)-representation Indε(V ) is irreducible. Proof. By its universal property, Indε(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 vλ′ is a highest weight vector inside M ε,+ λ at weight λ′ different from λ. By Lemma 2.5, it follows that∑ w∈W εω−wωq (−2wω,ρ)λ′−4wω = ∑ w∈W εω−wωq (−2wω,ρ)λ−4wω for all ω ∈ P+. Now if w = sαi1 · · · sαip in reduced form, with sα the reflection across the root α, we have ω − wω = p∑ t=1 sαi1 · · · sαit−1 (ω − sαitω), where each term is positive. It follows that we have∑ w∈WS q(−2wω,ρ)λ′−4wω = ∑ w∈WS q(−2wω,ρ)λ−4wω for all strictly dominant ω, where WS is the Coxeter group generated by reflections around simple roots αs with s ∈ S. Taking ω = ρ+ ωr with r /∈ S, we get (λ′−4ωr − λ−4ωr)C = 0 with C = ∑ w∈WS q(−2wρ,ρ)λ−4wρ > 0. Hence λ′ωr = λωr for all r /∈ S. We deduce that vλ′ ∈ Uq(tS)vλ, and so the image of vλ′ in Indε(V ) is zero. This implies that Indε(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 εt < 0 while εr ≥ 0 for r 6= t. This will in particular comprise the ‘symmetric Hermitian’ case. Theorem 2.7. Let ε be such that there is a unique simple root αt with εt < 0, while εr ∈ {0, 1} for r 6= t. Let λ ∈ CharR+(P ). Then V ε,+ λ is unitarizable if and only if λ4αr ∈ q 2N r for all r with εr 6= 1. Proof. By a same kind of limiting argument as in Proposition 2.3, the general case can be deduced from the case with εr = 1 for r 6= t. Suppose then that εt < 0 and εr = 1 for r 6= t. Write S = Φ+ \ {t}. We have the algebra automorphism φ : Uq(g; ε,+) → Uq(g) appearing in (1.1). By means of this isomorphism, we obtain a natural isomorphism M ε,+ λ ∼= M+,+ γ , where γ ∈ CharC(P ) is such that γ4αr = εrλ 4 αr for all r. In particular, γ2αt ∈ C \ R. It follows from [18, Proposition 5.13] that Indε(Vλ) is irreducible, where Vλ denotes the irreducible representation of Uq(q + S ) at highest weight λ. Hence the signatures of the Hermitian inner products on the Indε(Vλ) are constant as εt < 0 varies. Indeed, these spaces can be identified canonically with a fixed quotient of Uq(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 Indε(Vλ) for εt = 0 is positive definite. It follows that the Hermitian inner product on a weight space of Indε(Vλ) is positive for εt < 0 small. As the signature is constant, this holds for all εt < 0. � Representations of Interpolating QUE Algebras 9 3 Quantized homogeneous spaces Definition 3.1. For ε, η ∈ CharR(Q+), we denote by Uq(g; ε, η)fin the space of locally finite vectors in Uq(g; ε, η) with respect to the right adjoint action by Uq(g; η, η) (cf. Definition A.2), Uq(g; ε, η)fin = { x ∈ Uq(g; ε, η) ∣∣dim(x� Uq(g; η, η)) <∞ } . It is easily seen that the space Uq(g; ε, η)fin is a ∗-subalgebra of Uq(g; ε, η) (cf. [16, Corol- lary 2.3]), and in the following it will always be treated as a right Uq(g)-module by �. A similar definition of finUq(g; ε, η) can be made with respect to the left adjoint action of Uq(g; ε, ε), and the two resulting algebras Uq(g; ε, η)fin and finUq(g; ε, η) should in some sense be seen as dual to each other. For example, the Uq(g; ε,+)fin will lead to compact quantum homogeneous spaces, while the finUq(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 Uq(g; ε, η)fin are sufficiently large, as the next proposition shows, extending Lemma 2.4. Proposition 3.2. As a right Uq(g)-module, Uq(g; ε,+)fin is generated by the K−4ω with ω ∈ P+. The algebra generated by Uq(g; ε,+)fin and the K4 ωr equals the subalgebra of Uq(g; ε,+) generated by the K±4ωr and the KαrX ± r . Proof. Again, the proof of [16, Theorem 6.4] can be directly modified. � Note that for εr 6= 0 for all r, the above proposition follows more straightforwardly from [16] by a rescaling argument. The dependence of Uq(g; ε, η)fin on ε and η is weaker than for Uq(g; ε, η) itself. We consider a special case in the following lemma. Recall that A denotes the Cartan matrix. Lemma 3.3. Consider ε, η ∈ CharR\{0}(Q +), and write sgn(εr/ηr) = (−1)χr . If χ is in the range of Amod 2, then Uq(g; ε,+)fin ∼= Uq(g; η,+)fin as right Uq(g)-module ∗-algebras. Proof. Choose b ∈ CharC(P ) such that b4αr = ηr/εr. Then Ψε,η : Uq(g; ε,+)→ Uq(g; η,+), { X±r 7→ b−1αrX ± r , Kω 7→ bωKω is a unital �-equivariant algebra isomorphism. Hence Ψε,η induces a unital �-equivariant algebra isomorphism ψε,η : Uq(g; ε,+)fin → Uq(g; η,+)fin. As the K−4ω with ω ∈ P+ generate Uq(g; ε,+)fin as a module, ψε,η will be ∗-preserving if and only if b4αr ∈ R for all r. This can be realized if we can find cr ∈ {−1, 1} such that∏ s c asr s = sgn(ηr/εr), which is equivalent with the condition appearing in the statement of the lemma. � In particular, we find for example that Uq(sl(2m+1); ε,+)fin for m ∈ N0 is independent of the choice of ε ∈ CharR\{0}(Q +). On the other hand, Uq(sl(2); ε,+)fin are mutually non-isomorphic as ∗-algebras for ε ∈ {−1, 0, 1}, see [8]. Definition 3.4. Let ε, η ∈ CharR(Q+), λ ∈ CharR+(P ), and let V ε,η λ be the irreducible highest weight module of Uq(g; ε, η) at λ with associated representation πε,ηλ . We write Bλ(g; ε, η) = πε,ηλ (Uq(g; ε, η)) and Bfin λ (g; ε, η) = πε,ηλ (Uq(g; ε, η)fin). 10 K. De Commer Remark 3.5. The space Bfin λ (g; ε, η) is not defined as the space Bλ(g; ε, η)fin of locally finite �-elements in Bλ(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 Bλ(g; ε,+): Zr = πε,+λ ( K−4ωr ) , Xr = q1/2r (q−1r − qr)π ε,+ λ ( KαrK −4 ωr X + r ) , Yr = X∗r , Wr = πε,+λ ( K4 αrK −8 ωr ) , Tr = ( qr − q−1r )2 πε,+λ ( K2 αrK −4 ωr X + r X − r ) + εrq −1 r πε,+λ ( K4 αrK −4 ωr ) + qrπ ε,+ λ ( K−4ωr ) = ( qr − q−1r )2 πε,+λ ( K2 αrK −4 ωr X − r X + r ) + εrqrπ ε,+ λ ( K4 αrK −4 ωr ) + q−1r πε,+λ ( K−4ωr ) . The following commutation relations will be needed later on. Lemma 3.7. The elements Wr and Tr commute with Xr, Yr, Zr, Tr and Wr. Moreover, XrZr = q2rZrXr, YrZr = q−2r ZrYr and XrYr = −εrWr + qrTrZr − q2rZ2 r , YrXr = −εrWr + q−1r TrZr − q−2r Z2 r . We further have that Tr and Wr are invariant under �X±r and �Kω, while Xr �X+ r = 0, Yr �X+ r = −q1/2r ( q−1r + qr ) Zr + q1/2r Tr, Xr �Kω = q− (ω,αr) 2 Xr, Yr �Kω = q (ω,αr) 2 Xr, Xr �X−r = q−1/2r ( q−1r + qr ) Zr − q−1/2r Tr, Yr �X−r = 0 and Zr �X+ r = q1/2r Xr, Zr �Kω = 0, Zr �X−r = −q−1/2r Yr. Finally, all elements Xr, Yr, Zr, Tr, Wr are inside Bfin λ (g; ε,+). Proof. All these assertions follow from straightforward computations. As the Zr = πε,+λ (K−4ωr ) and Wr are in Bfin λ (g; ε,+) by Proposition 3.2, and the latter is �-stable, it follows from the above computations that also Xr, Yr and Tr are in Bfin λ (g; ε,+). � Proposition 3.8. The only �-invariant elements in Bfin λ (g; ε,+) are scalar multiples of the unit element. Proof. Assume that x ∈ Uq(g; ε,+)fin with πε,+λ (x) invariant. As Uq(g; ε,+)fin is a semi-simple right Uq(g)-module, we have an equivariant projection E of Uq(g; ε,+)fin onto the ∗-algebra of its invariant elements. The latter is simply the center Z (Uq(g; ε,+)) of Uq(g; ε,+), by Lemma A.3. As πε,+λ is �-equivariant by construction, we deduce that πε,+λ (x) = πε,+λ (E(x)). But the latter is a scalar. � Remark 3.9. An alternative proof consists in applying Schur’s lemma to the simple modu- le V ε,+ λ . Indeed, x ∈ Bfin λ (g; ε,+) is �-invariant if and only if it commutes with all πε,+λ (y) for y ∈ Uq(g; ε,+). As V ε,+ λ is simple, Schur’s lemma implies that the algebra of �-invariant elements in Bfin 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). Representations of Interpolating QUE Algebras 11 Proposition 3.10. Let (V, π) be a ∗-representation of Bfin 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 Bfin λ (g; ε,+) consists of locally finite elements, any b ∈ Bfin λ (g; ε,+) can be written as a finite linear combination of elements bi ∈ Bfin λ (g; ε,+) for which there exists a finite-di- mensional ∗-representation π of Uq(g) on a Hilbert space such that bi � h = ∑ j πij(h)bj for all h ∈ Uq(g), the πij being the matrix components with respect to some orthogonal basis. An easy computation shows that ∑ i b ∗ i bi 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(bi)ξ‖ ≤ C‖ξ‖. We deduce that the element π(b) is bounded. � Definition 3.11. A Bfin λ (g; ε,+)-module V is called a highest weight module if there exists a cyclic vector v ∈ V which is annihilated by all Xr and which is an eigenvector for all Zr 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 ∈ Bfin λ (g; ε,+). We aim to show that the Bfin λ (g; ε,+) have only a finite number of non-equivalent irreducible highest weight modules. Of course, each Bfin λ (g; ε,+) admits at least the highest weight mo- dule 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 Zr. Proposition 3.12. Each Bfin 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 εr ∈ {0, 1} for all r upon allowing λ ∈ CharC(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 Bfin λ (g; ε,+) is obtained by restriction of a Uq(g; ε,+)-module V ε,+ λ′ for some λ′ ∈ CharC(P ). As the center of Uq(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 εω−wωq −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 εr = 0, it follows as in the proof of Lemma 2.5 that∑ w∈WS q−2(wω,ρ)λ−4wω = ∑ w∈WS q−2(wω,ρ)λ′−4wω 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 WS-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 Zr = πε,+λ (K−4ωr ), this is sufficient to prove the proposition. � Remark that the above proof also gives the upper bound \|WS \| 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 Bfin λ (g; ε,+) on a Hilbert space H . If 0 is not in the point-spectrum of any of the Zr, then H is a (possibly infinite) direct sum of completions (Hk, πk) of unitarizable highest weight modules of Bfin λ (g; ε,+). 12 K. De Commer Proof. By a direct integral decomposition, and using Proposition 3.12, it is sufficient to show that any such irreducible ∗-representation π of Bfin λ (g; ε,+) on a Hilbert space H is the com- pletion of a highest weight module with invariant pre-Hilbert space structure. We then argue as in [27, Section 3]. Write χX for the characteristic function of a set. By assumption, there exists t ∈ Rl with tr 6= 0 for all r and Pt = χ∏ r[qrtr,tr] (π(Z1), . . . , π(Zl)) non-zero. Suppose now that r is such that π(Xr)Pt 6= 0. From the commutation relations between the Xr and the Zs, we deduce that P(t1,...,q −2 r tr,...,tl) 6= 0. As the π(Zr) are bounded, this process must necessarily stop. Hence we may choose t such that Pt 6= 0 but π(Xr)Pt = 0 for all r. Let V be the union of the images of the spectral projections of (Z1, . . . , Zl) corresponding to the ∏ r ( R \ (− 1 n , 1 n) ) with n ∈ N. As Bfin λ (g; ε,+) is spanned by elements which skew-commute with the Zr, it follows that V is a Bfin λ (g; ε,+)-module on which the π(Zr) are invertible linear maps. This entails that the restriction of π to V can be extended to a representation π̃ of Bfin(g; ε,+)ext, the sub∗-algebra of B(g; ε,+) generated by Xr, Yr and the Z±1r (which contains Bfin 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 ∈ PtH . Suppose that ξ were not in the pure point spectrum of some π(Zr). Then we can find qrtr < a < tr such that χ[qrtr,a](π(Zr))ξ 6= 0 6= χ(a,tr](π(Zr))ξ. However, [q1t1, t1] × · · · × [qrtr, a] × · · · × [qltl, tl] ∩ ∏ s[q 2ks+1 s ts, q 2ks s ts] = ∅ for all ks ∈ N with at least one ks > 0. From the commutation relations between the Ys and Zs′ , and the fact that π(Xs)ξ = 0 for all s, we deduce that χ[qrtr,a](π(Zr))ξ is orthogonal to the Bfin λ (g; ε,+)ext-module spanned by χ(a,tr](π(Zr))ξ. As π is irreducible, this would entail χ[qrtr,a](π(Zr))ξ = 0. Having arrived at a contradiction, we conclude that ξ is a joint eigenvector of all π(Zr). As ξ is annihilated by all π(Xr) and is a joint eigenvector of all π(Zr), 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 Bfin 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 πu : 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(Gq +) to be the Hopf ∗-algebra inside the dual of Uq(g) which is spanned by the matrix coefficients of finite-dimensional highest weight representations of Uq(g) associated to positive characters. We define αε,+λ : Bfin λ (g; ε,+)→ Pol(Gq +)⊗Bfin λ (g; ε,+) as the comodule ∗-algebra structure dual to the module ∗-algebra structure � by Uq(g). Note that the latter definition makes sense, since Bfin λ (g; ε,+) is integrable as a right Uq(g)- module. It is known [26] that Pol(Gq +) admits a universal C∗-algebraic envelope C(Gq +), which becomes a compact quantum group in the sense of [32]. We will denote by ϕGq+ the invariant state on C(Gq +), which is faithful by co-amenability of Gq +. Lemma 3.16. Assume that Bfin λ (g; ε,+) admits at least one ∗-representation on a Hilbert space. Then Bfin λ (g; ε,+) admits a C∗-algebraic envelope Cλ(g; ε,+). Representations of Interpolating QUE Algebras 13 Proof. The universal C∗-algebraic envelope of Bfin λ (g; ε,+) exists for precisely the same rea- son as in Proposition 3.10, since for any element b ∈ Bfin λ (g; ε,+) there exists a universal constant Cb such that ‖π(b)‖ ≤ C for all ∗-representations π of Bfin λ (g; ε,+) by bounded operators on a Hilbert space. As Bfin λ (g; ε,+) admits at least one ∗-representation, we have Cλ(g; ε,+) 6= 0. � Remark 3.17. If V ε,+ λ is unitarizable, it is of course clear that we get a faithful map from Bfin λ (g; ε,+) into Cλ(g; ε,+). Lemma 3.18. Let Cλ(g; ε,+) be the universal C∗-envelope of Bfin λ (g; ε,+), whenever it exists. Then αε,+λ induces a C∗-algebraic coaction by C(Gq +) on Cλ(g; ε,+). Proof. The map αε,+λ gives a C∗-representation of Bfin λ (g; ε,+) into C(Gq +)⊗Cλ(g; ε,+), which hence factors over Cλ(g; ε,+). It is straightforward to argue that this is a C∗-algebraic coac- tion. � As Bfin λ (g; ε,+) only had scalar multiples of the unit as invariants, it follows that the coac- tion αε,+λ on Cλ(g; ε,+) is ergodic [3], i.e. if αε,+λ (x) = 1 ⊗ x, then x ∈ C1. We will write ϕε,+λ for the unique invariant state on Cλ(g; ε,+), so( ϕGq+ ⊗ ι ) αε,+λ (x) = ϕε,+λ (x)1, ∀x ∈ Cλ(g; ε,+). As ϕGq+ is faithful, also ϕε,+λ is faithful. Notation 3.19. We write θε,+λ,reg for the GNS-representation of (Cλ(g; ε,+), ϕε,+λ ), andWλ(g; ε,+) for the von Neumann algebraic completion of Cλ(g; ε,+) in this GNS-representation. From Lemma 3.16, it follows that a C∗-algebraic envelope of Bfin λ (g; ε,+) exists if V ε,+ λ is unitarizable. In this case, we can say something explicit about Wλ(g; ε,+). Theorem 3.20. Assume that V ε,+ λ is unitarizable. Then Wλ(g; ε,+) is a finite direct sum of type I factors. 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 . Proof. As ω is faithful, the bicommutant π(A)′′ is faithfully represented on the GNS-space L 2(π(A), ω). The unitary U : L 2(A,ϕ)→ L 2(π(A), ω) induced by π then provides an isomor- phism M → π(A)′′ extending π. � Proof of Theorem 3.20. As mentioned, the C∗-algebraic envelope Cλ(g; ε,+) certainly exists. To prove the remaining part of the theorem, we first make some preparations. Recall that the dual of Uq(sl(2,C)) can be identified with Pol(SUq(2)), Woronowicz’s twisted quantum SU(2)-group [33]. It is well-known that its von Neumann algebraic envelope L∞(SUq(2)) is isomorphic to B(l2(N)) ⊗ L (Z), and as such admits a faithful representation on the Hilbert space H+ = l2(N)⊗ l2(Z) (cf. [26, 29]). More generally, write Uqr(su(2)) for the sub-Hopf-∗-algebra generated by the X±r and K±1αr inside Uq(g). By duality, one obtains a surjective ∗-homomorphism γr : Pol(Gq +)→ Pol(SUqr(2)). This induces a ∗-representation of Pol(Gq +) on H+, which we will denote by the same symbol γr. 14 K. De Commer Suppose now that t = (r1, . . . , rn) is an ordered n-tuple of elements in I. Then we obtain a ∗-representation of Pol(Gq+) on H ⊗n + by means of the ∗-homomorphism γt = (γr1 ⊗ · · · ⊗ γrn) ◦∆ (n) Gq+ , where ∆ (n) Gq+ denotes the n-fold coproduct. Let now t0 = (r1, . . . , rN ) be such that w0 = sr1 · · · srN is a reduced expression for the longest element in the Weyl group of g. By [29], we know that ϕGq+ can be realized as ω ◦ γt0 for some faithful normal state ω ∈ B ( H ⊗N + ) ∗. By Lemma 3.21, this implies that γt0 can be extended to a faithful normal ∗-representation of L∞(Gq+). Let us turn now to the proof of the theorem. The main step is to prove that the point- spectrum of θε,+λ,reg ( l∏ r=1 Zr ) does not contain zero. Indeed, if this is the case, then we can use Proposition 3.13 to conclude that θε,+λ,reg decomposes into a direct integral of highest weight modules for Bfin λ (g; ε,+). It then follows that Wλ(g; ε,+) is simply ⊕mk=1B(Hk) with Hk the Hilbert space completion of the highest weight modules which appear in θε,+λ,reg. To show that θε,+λ,reg ( l∏ r=1 Zr ) does not contain zero in its pointspectrum, it is sufficient to show that the operator (γt0⊗θ ε,+ λ ) ( αε,+λ ( l∏ r=1 Zr )) does not contain zero in its pointspectrum, where θε,+λ is the ∗-representation of Bfin λ (g; ε,+) on the Hilbert space completion H ε,+ λ of V ε,+ λ . Indeed, the invariant state ϕε,+λ = (ϕGq+⊗id)αε,+λ can be extended to a faithful normal functional on B(H ε,+ λ ⊗H+ ⊗N ), which implies, again by Lemma 3.21, that (γt0 ⊗ θ ε,+ λ )αε,+λ extends to a faithful normal ∗-representation of Wλ(g; ε,+). Finally, to show that (θε,+λ ⊗γt0) ( αε,+λ ( l∏ r=1 Zr )) 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 Bfin λ (g; ε,+) with an in- variant pre-Hilbert space structure, and let H be the completion of V . Fix r ∈ I, and put πr = (π ⊗ γr)αε,+λ . Then πr ( l∏ s=1 Zs ) does not contain 0 in its point spectrum. Proof. It is easy to see that πr(Zs) = 1⊗ π(Zs) for r 6= s, so that none of these operators have zero in their point-spectrum. Let now Ar be the sub-∗-algebra of Bfin λ (g; ε,+) generated by Zr, Tr, Xr, Yr, Wr, see Lemma 3.7. By that lemma, Ar is stable under the right action by Uqr(su(2)), and Wr and Tr are invariants in the center of Ar. By invariance, πr(Wr) = 1 ⊗ π(Wr) and πr(Tr) = 1 ⊗ π(Tr) are bounded self-adjoint operators. Hence, to investigate the spectrum of πr(Zr), we may by disintegration treat the above operators as scalars, say wr and tr. Denote the resulting quotient of Ar by Ar(wr, tr). From Lemma 3.7, we find commutation relations between the generators Xr, Yr and Zr of Ar(wr, tr), as well as the resulting action of Uqr(su(2)). It follows that Ar(wr, tr) is an equivariant quotient of a generalized Podleś sphere S2 qr,τ for SUqr(2) [27], for some τ depending on tr and wr. Moreover, as ϕSUqr (2) can be realized on the Hilbert space H+, the von Neumann algebraic envelope of Ar(wr, tr) will be isomorphic to the von Neumann algebraic envelope of S2 qr,τ , which is equal to Mn(C), B(l2(N)) or B(l2(N)) ⊕ B(l2(N)), depending on whether εrwr is positive, zero or negative (cf. [27]). In any case, the corresponding image of Zr will not contain 0 in its point-spectrum. � Representations of Interpolating QUE Algebras 15 4 More on the ε ∈ Char{0,1}(Q +)-case The case of the non-standard Podleś spheres already shows that Wλ(g; ε,+) is in general not a factor. However, for ε ∈ Char{0,1}(Q +) and λ ∈ CharR+(P ) such that Wλ(g; ε,+) is well- defined, we show that Wλ(g; ε,+) does become a type I-factor, and we can then also say some- thing more about the invariant integral ϕε,+λ on Wλ(g; ε,+). Proposition 4.1. Let ε ∈ Char{0,1}(Q +), λ ∈ CharR+(P ), and suppose V ε,+ λ is unitarizable with completion H ε,+ λ . Identify Bfin λ (g; ε,+) ⊆ B(H ε,+ λ ). Then this inclusion completes to a natural identification Wλ(g; ε,+) ∼= B(H ε,+ λ ). 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 Ar(wr, tr) appearing in the proof of Lemma 3.22 can only be matrix algebras or standard Podleś spheres. As the Zr in the von Neumann algebraic completion of these algebras are always positive operators, it follows by induction that the components appearing in Wλ(g; ε,+) arise from restrictions of highest weight modules V ε,+ λ′ of Uq(g; ε,+) with λ′ ∈ CharR+(P ). Our aim is to show that necessarily λ′ = λ. Let S be the set of r with εr = 1. Suppose that λ′ ∈ CharR+(P ) is such that the repre- sentation of Uq(g; ε,+) on V ε,+ λ′ factors over Bfin λ (g; ε,+). Suppose that V ε,+ λ′ then admits an invariant pre-Hilbert space structure as a Bfin λ (g; ε,+)-module, hence as a Uq(g; ε,+)-module by Proposition 3.2. From the proof of Proposition 3.12, we deduce that there exists w ∈ WS such that q(−2wω,ρ)λ−4wω = q(−2ω,ρ)λ′−4ω for all ω ∈ P . In particular, λωr = λ′ωr for r /∈ S. On the other hand, let Ũq(kss) be the subalgebra of Uq(g; ε,+) generated by the X±r and K±1αr with r ∈ S. Let Ṽ ε,+ λ be the Ũq(kss)-module spanned by vε,+λ , and similarly for V ε,+ λ′ . Then by Proposition 2.6, these are irreducible highest weight modules for Uq(kss) associated to the restrictions of λ and λ′ to the root lattice QS of kss. But as V ε,+ λ′ admits an invariant pre- Hilbert space structure (and kss is compact), it is necessarily finite-dimensional. However, as the restriction of λ′ lies in the WS-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 QS coincide. Combined with the observation at the end of the previous paragraph, this forces λ = λ′ inside CharR+(P ). � Proposition 4.2. Under the assumptions of Proposition 4.1, the invariant state on Wλ(g; ε,+) is given by ϕε,+λ (x) = Tr(xZρ) Tr(Zρ) , where Zρ = l∏ r=1 Zr. Proof. Consider the projection Eε,+ of Uq(g; ε,+)fin onto its direct summand Z (Uq(g; ε,+)), the space of �-invariants. For εr > 0 for all r, it follows from [16, Chapter 7] that Eε,+(xy) = Eε,+(yσ(x)) for all x, y ∈ Uq(g; ε,+)fin, where σ(x) = K−4ρxK −1 −4ρ. By continuity, this then holds for all ε ∈ CharR(Q+). Clearly σ induces an automorphism of Bfin λ (g; ε,+), which we will denote by the same symbol. As ϕε,+λ factors over Eε,+, we have as well that ϕε,+λ (xy) = ϕελ(yσ(x)). However, by general theory [3] we know that the modular automorphism group σt of ϕε,+λ leaves Bfin λ (g; ε,+) invariant and is diagonalizable (and hence analytic) on it. From the above, we can hence conclude that σt = Ad(Zitρ ) (where we use that Zρ is a positive operator by Proposition 4.1). As Wλ(g; ε,+) is a type I-factor, again by Proposition 4.1, we know that ϕε,+λ is completely determined by its modular automorphism group up to a scalar. Hence we obtain our expression for ϕε,+λ as in the statement of the proposition. � 16 K. De Commer 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 {Hij \| 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 ∆k ij : Hij → Hik ⊗Hkj , h 7→ h(1)ik ⊗ h(2)kj . We call ( {Hij}, {∆k ij} ) a connected cogroupoid over the index set I if the following conditions are satisfied: • (Connectedness) None of the Hij are the zero algebra. • (Coassociativity) For each quadruple i, j, k, l of indices, we have( id⊗∆k jl ) ∆j il = ( ∆j ik ⊗ id ) ∆k il, • (Counits) There exist unital ∗-homomorphisms εi : Hii → C such that, for all indices i, j, (εi ⊗ id)∆i ij = idHij = (id⊗εj)∆j ij , • (Antipodes) There exist anti-homomorphisms Sij : Hij → Hji such that, for all indices i, j and all h ∈ Hii, we have Sij(h(1)ij)h(2)ji = εi(h) = h(1)ijSji(h(2)ji). As for Hopf ∗-algebras, it is easy to show that the Sij are unique, and that Sji(Sij(h)∗)∗ = h for each h ∈ Hij . Note that each ( Hii,∆ i ii ) defines a Hopf ∗-algebra. Definition A.2. Let ( Hij ,∆ k ij ) be a cogroupoid. The right adjoint action (or Miyashita–Ulbrich action) � of Hjj on Hij is given by x� h = Sji(h(1)ji)xh(2)ij . One easily proves that � defines a right Hjj-module ∗-algebra structure on Hij . The com- patibility with the ∗-structure means that (x� h)∗ = x∗ � Sjj(h)∗. Lemma A.3. The space of �-invariant elements in Hij coincides with the center of Hij. Proof. (Cf. [16, Lemma 2.4].) If x ∈ Hij is in the center, clearly x�h = εj(h)x for all h ∈ Hjj , by definition of the antipode. Conversely, if Sji(h(1)ji)xh(2)ij = εj(h)x for all h ∈ Hjj , we have for y ∈ Hij that xy = y(1)ijSji(y(2)ji)xy(3)ij = yx. � Suppose now that B is a unital ∗-algebra, and that for some i, j we have a unital ∗-homo- morphism π : Hij → B. As the right action of Hjj on Hij is inner, it descends to a right action on B, b� h = π(Sji(h(1)ji))bπ(h(2)ij). Note that the central elements in B are invariant for the action. A particular case of cogroupoid can be constructed from a Hopf algebra (H,∆) together with a collection of real 2-cocycle functionals {ωi \| i ∈ I} on it. Here we mean by real 2- cocycle functional an element ω ∈ (H ⊗ H)∗ which is convolution invertible and such that ω(1, h) = ε(h) = ω(h, 1) for all h ∈ H, while ω(h∗, k∗) = ω(k, h) and ω(h(1), k(1))ω(h(2)k(2), l) = ω(k(1), l(1))ω(h, k(2)l(2)), ∀h, k, l ∈ H. Representations of Interpolating QUE Algebras 17 Let us write Hij for the vector space H with the new multiplication mij : H ⊗H → H, h⊗ k 7→ ωi(h(1), k(1))h(2)k(2)ω −1 j (h(3), k(3)), and write ∆k ij for the given coproduct ∆ seen as a map Hij → Hik ⊗Hkj . Then the (Hij ,∆ k ij) form a connected cogroupoid. 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 {hr} ⊆ h for the basis dual to {ωr}, and write hR for its real span, which we may identify with the real dual of h∗R. We write gc ⊆ g for the compact real form of g, and † for the corresponding anti-linear anti-automorphism such that x† = −x for x ∈ gc. 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 ] = hr for all αr ∈ Φ+. Fix ε ∈ CharR(Q+). We can define on g the linear maps S±ε : g→ g : { X±α 7→ εαX ± α , α ∈ ∆+, z 7→ z, z ∈ h + n∓. Definition B.1. We define gε to be the complex vector space {(S+ ε (z), S−ε (z)) \| z ∈ g} ⊆ g⊕ g. Proposition B.2. The vector space gε is a Lie ∗-subalgebra of the direct sum Lie algebra g⊕ g equipped with the involution (w, z)∗ = (z†, w†). Moreover, dimC(gε) = dimC(g). Proof. From the definition, we see that gε is the linear space generated by elements of the form (εαX + α , X + α ), (X−α , εαX − α ) and (hr, hr). Accordingly, gε has dimension dimC(g) and inherits the ∗-operation from g⊕ g. Since εα+β = εαεβ whenever α, β and α+ β are positive roots, we find that gε is closed under the bracket operation. � In the following, we consider gε with its ∗-operation inherited from g⊕ g. Remark B.3. By rescaling the X±α , we see that gε ∼= gη as Lie ∗-algebras whenever εr = λrηr for certain λr > 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 gRε = {z ∈ gε \| z∗ = −z}. Hence gRε is a real Lie algebra with gε as its complexification. We can also realize gRε more conveniently inside g as follows. Proposition B.5. Write X(ε) α = X+ α − εαX−α , Y (ε) α = i ( X+ α + εαX − α ) as elements in g. Consider the R-linear span of the X (ε) α , Y (ε) α and ihr. Then this space is closed under the Lie bracket of g, and forms a real Lie algebra isomorphic to gRε . 18 K. De Commer Proof. As a real Lie algebra, one can embed g inside the direct sum g⊕g by means of the map g→ g⊕ g, z 7→ ( z,−z† ) . Under this identification, it is immediately verified that the elements in the statement of the proposition get sent to a basis of gRε . � Proposition B.6. Suppose εr 6= 0 for all r. Then gRε is a semi-simple real Lie algebra. Proof. In this case, the projection onto the first coordinate of g⊕ g provides an isomorphism between gε and g. � Hence if ε ∈ Char{−1,1}(Q +), the Lie algebra gRε is the real form of g corresponding to the involution (X+ r )∗ = εrX − r , h∗r = hr. Accordingly, gRε is a real Lie algebra of equal rank, meaning that we can take a Cartan decom- position gR = t⊕p with t compact and ihR ⊆ t. Conversely, it is easy to see that any equal rank semi-simple real Lie algebra can be realized as some gRε . We recall that the equal rank semi- simple real Lie algebras are precisely those which admit an irreducible unitary representation of discrete type. As an example, let us present the simple equal rank real Lie algebras of type (A) (see [14, Chapter X]). For r ∈ {0, . . . , l}, choose σr ∈ {±1} such that εr = σr−1σr for r ∈ {1, . . . , l}. Then gRε is always of type (AIII), and ε corresponds to su(p, l + 1 − p) with p the number of negative σr. This precise correspondence between the ε and the various real Lie algebras is easy to determine explicitly from the standard descriptions of the Cartan decompositions. More generally, we obtain from Proposition B.5 the following characterization of which Lie algebras appear as some gε. Corollary B.7. Let p = l ⊕ u be a parabolic Lie subalgebra of g with Levi factor l and largest nilpotent ideal u. Let lR be a real form of l which is of equal rank on each simple summand and compact on the center. Then lR ⊕ u ∼= gRε as real Lie algebras for some ε ∈ CharR(Q+), and all gRε arise in this way. Our next step is to present gε by means of generators and relations. Definition B.8. Fix ε ∈ CharR(Q+). We define g̃ε as a universal Lie algebra in the following way. A set of generators is given by a triple of elements X±r and Hr for each simple root αr. The relations can be grouped into four parts, which we label as ‘(H)-condition’, ‘(T)orus action’, ‘(S)erre relations’ and ‘(C)oupling conditions’: for all r, s ∈ I, (H) [Hr, Hs] = 0, (T) [Hr, X ± s ] = ±arsX±s , (S) ad(X±r )1−ars(X±s ) = 0 when r 6= s, (C)ε [X+ r , X − s ] = δrsεrHr. It is immediate that g̃ε can be endowed with a ∗-operation such that (X+ r )∗ = X−r and H∗r = Hr. Proposition B.9. The Lie ∗-algebras g̃ε and gε are isomorphic. Proof. It is straightforward to verify that there is a unique Lie ∗-algebra homomorphism φ : g̃ε → gε ⊆ g⊕g such that φ(X+ r ) = (εrX + r , X + r ), φ(X−r ) = (X−r , εrX − r ) and φ(Hr) = (hr, hr). It is obviously surjective. On the other hand, it is easy to see by induction that each element of g̃ε can be written in the form x+ y + z with x in the Lie algebra generated by the X+ r ’s, z in the Lie algebra generated by the X−r ’s, and y in the linear span of the Hr. Since n± are universal with respect to the relations (S), we find that dim(g̃ε) ≤ dim(gε), hence φ is bijective. � Representations of Interpolating QUE Algebras 19 Acknowledgments It is a pleasure to thank the following people for discussions on topics related to the subject of this paper: J. Bichon, P. Bieliavsky, H.P. Jakobsen, E. Koelink, S. Kolb, U. Krähmer and S. Neshveyev. References [1] Bichon J., Hopf–Galois systems, J. Algebra 264 (2003), 565–581, math.QA/0204348. [2] Bichon J., Hopf–Galois objects and cogroupoids, arXiv:1006.3014. [3] Boca F.P., Ergodic actions of compact matrix pseudogroups on C∗-algebras, Astérisque 232 (1995), 93–109. [4] Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1995. [5] De Commer K., Comonoidal W∗-Morita equivalence for von Neumann bialgebras, J. Noncommut. Geom. 5 (2011), 547–571, arXiv:1004.0824. [6] De Commer K., On a correspondence between SUq(2), Ẽq(2) and S̃Uq(1, 1), Comm. Math. Phys. 304 (2011), 187–228, arXiv:1004.4307. [7] De Commer K., On a Morita equivalence between the duals of quantum SU(2) and quantum Ẽ(2), Adv. Math. 229 (2012), 1047–1079, arXiv:0912.4350. [8] De Commer K., On the construction of quantum homogeneous spaces from ∗-Galois objects, Algebr. Repre- sent. Theory 15 (2012), 795–815, arXiv:1001.2153. [9] De Concini C., Kac V.G., Representations of quantum groups at roots of 1, in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math., Vol. 92, Birkhäuser Boston, Boston, MA, 1990, 471–506. [10] Donin J., Mudrov A., Explicit equivariant quantization on coadjoint orbits of GL(n,C), Lett. Math. Phys. 62 (2002), 17–32, math.QA/0206049. [11] Enock M., Morita equivalence of measured quantum groupoids. Application to deformation of measured quantum groupoids by 2-cocycles, in Operator Algebras and Quantum Groups, Banach Center Publ., Vol. 98, Editors W. Pusz, P.M. Soltan, Polish Acad. Sci. Inst. Math., Warsaw, 2012, 107–198, arXiv:1106.1018. [12] Gilmore R., Lie groups, physics, and geometry. An introduction for physicists, engineers and chemists, Cambridge University Press, Cambridge, 2008. [13] Hayashi T., Face algebras. I. A generalization of quantum group theory, J. Math. Soc. Japan 50 (1998), 293–315. [14] Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, Vol. 80, Academic Press Inc., New York, 1978. [15] Humphreys J.E., Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, Vol. 94, American Mathematical Society, Providence, RI, 2008. [16] Joseph A., Letzter G., Local finiteness of the adjoint action for quantized enveloping algebras, J. Algebra 153 (1992), 289–318. [17] Joseph A., Letzter G., Separation of variables for quantized enveloping algebras, Amer. J. Math. 116 (1994), 127–177. [18] Joseph A., Todorić D., On the quantum KPRV determinants for semisimple and affine Lie algebras, Algebr. Represent. Theory 5 (2002), 57–99. [19] Karolinsky E., Stolin A., Tarasov V., Irreducible highest weight modules and equivariant quantization, Adv. Math. 211 (2007), 266–283, math.QA/0507348. [20] Kashiwara M., On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516. [21] Kassel C., Schneider H.J., Homotopy theory of Hopf Galois extensions, Ann. Inst. Fourier (Grenoble) 55 (2005), 2521–2550, math.QA/0402034. [22] Knapp A.W., Representation theory of semisimple groups. An overview based on examples, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001. http://dx.doi.org/10.1016/S0021-8693(03)00140-6 http://arxiv.org/abs/math.QA/0204348 http://arxiv.org/abs/1006.3014 http://dx.doi.org/10.4171/JNCG/86 http://arxiv.org/abs/1004.0824 http://dx.doi.org/10.1007/s00220-011-1208-y http://arxiv.org/abs/1004.4307 http://dx.doi.org/10.1016/j.aim.2011.10.011 http://dx.doi.org/10.1016/j.aim.2011.10.011 http://arxiv.org/abs/0912.4350 http://dx.doi.org/10.1007/s10468-011-9265-7 http://dx.doi.org/10.1007/s10468-011-9265-7 http://arxiv.org/abs/1001.2153 http://dx.doi.org/10.1023/A:1021677725539 http://arxiv.org/abs/math.QA/0206049 http://arxiv.org/abs/1106.1018 http://dx.doi.org/10.1017/CBO9780511791390 http://dx.doi.org/10.2969/jmsj/05020293 http://dx.doi.org/10.1016/0021-8693(92)90157-H http://dx.doi.org/10.2307/2374984 http://dx.doi.org/10.1023/A:1014410819083 http://dx.doi.org/10.1023/A:1014410819083 http://dx.doi.org/10.1016/j.aim.2006.08.004 http://dx.doi.org/10.1016/j.aim.2006.08.004 http://arxiv.org/abs/math.QA/0507348 http://dx.doi.org/10.1215/S0012-7094-91-06321-0 http://arxiv.org/abs/math.QA/0402034 20 K. De Commer [23] Korogodsky L.I., Representations of quantum algebras arising from non-compact quantum groups: Quan- tum orbit method and super-tensor products, Ph.D. Thesis, Massachusetts Institute of Technology, 1996, available at http://dspace.mit.edu/handle/1721.1/39076. [24] Letzter G., Quantum symmetric pairs and their zonal spherical functions, Transform. Groups 8 (2003), 261–292, math.QA/0204103. [25] Letzter G., Quantum zonal spherical functions and Macdonald polynomials, Adv. Math. 189 (2004), 88–147, math.QA/0210447. [26] Levendorskĭı S., Soibelman Y., Algebras of functions on compact quantum groups, Schubert cells and quan- tum tori, Comm. Math. Phys. 139 (1991), 141–170. [27] Masuda T., Nakagami Y., Watanabe J., Noncommutative differential geometry on the quantum two sphere of Podleś. I. An algebraic viewpoint, K-Theory 5 (1991), 151–175. [28] Mudrov A., Quantum conjugacy classes of simple matrix groups, Comm. Math. Phys. 272 (2007), 635–660, math.QA/0412538. [29] Reshetikhin N., Yakimov M., Quantum invariant measures, Comm. Math. Phys. 224 (2001), 399–426, math.QA/0101048. [30] Rosso M., Groupes quantiques, représentations linéaires et applications, Ph.D. Thesis, Universite de Paris VII, 1990. [31] Vaksman L.L., Quantum bounded symmetric domains, Translations of Mathematical Monographs, Vol. 238, American Mathematical Society, Providence, RI, 2010. [32] Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665. [33] Woronowicz S.L., Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117–181. http://dspace.mit.edu/handle/1721.1/39076 http://dx.doi.org/10.1007/s00031-003-0719-9 http://arxiv.org/abs/math.QA/0204103 http://dx.doi.org/10.1016/j.aim.2003.11.007 http://arxiv.org/abs/math.QA/0210447 http://dx.doi.org/10.1007/BF02102732 http://dx.doi.org/10.1007/BF01254546 http://dx.doi.org/10.1007/s00220-007-0222-6 http://arxiv.org/abs/math.QA/0412538 http://dx.doi.org/10.1007/PL00005587 http://arxiv.org/abs/math.QA/0101048 http://dx.doi.org/10.1007/BF01219077 http://dx.doi.org/10.2977/prims/1195176848 http://dx.doi.org/10.2977/prims/1195176848 Introduction 1 Two-parameter deformations of quantized enveloping algebras 2 Verma module theory 3 Quantized homogeneous spaces 4 More on the `39`42`"613A``45`47`"603AChar{0,1}(Q+)-case A Cogroupoids B Continuous one-parameter families of Lie algebras References

Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

Institution

Ähnliche Einträge