A locally compact quantum group of triangular matrices

We construct a one parameter deformation of the group of 2×2 upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a comple...

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Datum:	2008
Hauptverfasser:	Fima, P., Vainerman, L.
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Veröffentlicht:	Інститут математики НАН України 2008
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spelling	nasplib_isofts_kiev_ua-123456789-1644942025-02-09T23:39:07Z A locally compact quantum group of triangular matrices Локально компактна квантова група трикутних матриць Fima, P. Vainerman, L. Статті We construct a one parameter deformation of the group of 2×2 upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual C∗-algebra and the dual comultiplication. Побудовано однопараметричну деформацію групи верхніх трикутних матриць розміру 2 × 2 із детермінантом 1 з використанням конструкції скруту. Цікавою рисою цього нового прикладу локально компактної квантової групи є нетривіальна деформація міри Хаара. Наведено також повний опис дуальної C*-алгебри та дуальної комультиплікації. 2008 Article A locally compact quantum group of triangular matrices / P. Fima, L. Vainerman // Український математичний журнал. — 2008. — Т. 60, № 4. — С. 564–576. — Бібліогр.: 16 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164494 517.5 en Український математичний журнал application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Статті Статті
spellingShingle	Статті Статті Fima, P. Vainerman, L. A locally compact quantum group of triangular matrices Український математичний журнал
description	We construct a one parameter deformation of the group of 2×2 upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual C∗-algebra and the dual comultiplication.
format	Article
author	Fima, P. Vainerman, L.
author_facet	Fima, P. Vainerman, L.
author_sort	Fima, P.
title	A locally compact quantum group of triangular matrices
title_short	A locally compact quantum group of triangular matrices
title_full	A locally compact quantum group of triangular matrices
title_fullStr	A locally compact quantum group of triangular matrices
title_full_unstemmed	A locally compact quantum group of triangular matrices
title_sort	locally compact quantum group of triangular matrices
publisher	Інститут математики НАН України
publishDate	2008
topic_facet	Статті
url	https://nasplib.isofts.kiev.ua/handle/123456789/164494
citation_txt	A locally compact quantum group of triangular matrices / P. Fima, L. Vainerman // Український математичний журнал. — 2008. — Т. 60, № 4. — С. 564–576. — Бібліогр.: 16 назв. — англ.
series	Український математичний журнал
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first_indexed	2025-12-01T20:07:07Z
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fulltext	UDC 517.5 P. Fima (Univ. Franche-Comté, France), L. Vainerman (Univ. Caen, France) A LOCALLY COMPACT QUANTUM GROUP OF TRIANGULAR MATRICES ЛОКАЛЬНО КОМПАКТНА КВАНТОВА ГРУПА ТРИКУТНИХ МАТРИЦЬ We construct a one parameter deformation of the group of 2×2 upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual C∗-algebra and the dual comultiplication. Побудовано однопараметричну деформацiю групи верхнiх трикутних матриць розмiру 2 × 2 iз детермiнантом 1 з використанням конструкцiї скруту. Цiкавою рисою цього нового прикладу локально компактної квантової групи є нетривiальна деформацiя мiри Хаара. Наведено також повний опис дуальної C∗-алгебри та дуальної комультиплiкацiї. 1. Introduction. In [1, 2], M. Enock and the second author proposed a systematic approach to the construction of non-trivial Kac algebras by twisting. To illustrate it, consider a cocommutative Kac algebra structure on the group von Neumann algebra M = L(G) of a non commutative locally compact (l.c.) group G with comultiplication ∆(λg) = λg⊗λg (here λg is the left translation by g ∈ G). Let us define on M another, “twisted”, comultiplication ∆Ω(·) = Ω∆(·)Ω∗, where Ω is a unitary from M ⊗ M verifying certain 2-cocycle condition, and construct in this way new, non cocommutative, Kac algebra structure on M. In order to find such an Ω, let us, following to M. Rieffel [3] and M. Landstad [4], take an inclusion α : L∞(K̂) → M, where K̂ is the dual to some abelian subgroup K of G such that δ\|K = 1, where δ(·) is the module of G. Then, one lifts a usual 2-cocycle Ψ of K̂ : Ω = (α⊗α)Ψ. The main result of [1, 2] is that the integral by the Haar measure of G gives also the Haar measure of the deformed object. Recently P. Kasprzak studied the deformation of l.c. groups by twisting in [5], and also in this case the Haar measure was not deformed. In [6], the authors extended the twisting construction in order to cover the case of non-trivial deformation of the Haar measure. The aim of the present paper is to illustrate this construction on a concrete example and to compute explicitly all the ingredients of the twisted quantum group including the dual C∗-algebra and the dual comultiplication. We twist the group von Neumann algebra L(G) of the group G of 2×2 upper triangular matrices with determinant 1 using the abelian subgroup K = C∗ of diagonal matrices of G and a one parameter family of bicharacters on K. In this case, the subgroup K is not included in the kernel of the modular function of G, this is why the Haar measure is deformed. We compute the new Haar measure and show that the dual C∗-algebra is generated by 2 normal operators α̂ and β̂ such that α̂β̂ = β̂α̂, α̂β̂∗ = qβ̂∗α̂, where q > 0. Moreover, the comultiplication ∆̂ is given by c© P. FIMA, L. VAINERMAN, 2008 564 ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 A LOCALLY COMPACT QUANTUM GROUP OF TRIANGULAR MATRICES 565 ∆̂t(α̂) = α̂⊗ α̂, ∆̂t(β̂) = α̂⊗ β̂+̇β̂ ⊗ α̂−1, where +̇ means the closure of the sum of two operators. This paper in organized as follows. In Section 2 we recall some basic definitions and results. In Section 3 we present in detail our example computing all the ingredients associated. This example is inspired by [5], but an important difference is that in the present example the Haar measure is deformed in a non trivial way. Finally, we collect some useful results in the Appendix. 2. Preliminaries. 2.1. Notations. Let B(H) be the algebra of all bounded linear operators on a Hilbert space H, ⊗ the tensor product of Hilbert spaces, von Neumann algebras or minimal tensor product of C∗-algebras, and Σ (resp., σ) the flip map on it. If H,K and L are Hilbert spaces and X ∈ B(H ⊗ L) (resp., X ∈ B(H ⊗ K), X ∈ ∈ B(K⊗L)), we denote by X13 (resp., X12, X23) the operator (1⊗Σ∗)(X⊗1)(1⊗Σ) (resp., X ⊗ 1, 1 ⊗ X) defined on H ⊗ K ⊗ L. For any subset X of a Banach space E, we denote by 〈X〉 the vector space generated by X and [X] the closed vector space generated by X. All l.c. groups considered in this paper are supposed to be second countable, all Hilbert spaces are separable and all von Neumann algebras have separable preduals. Given a normal semi-finite faithful (n.s.f.) weight θ on a von Neumann algebra M (see [7]), we denote M+ θ = { x ∈ M+ \| θ(x) < +∞ } , Nθ = { x ∈ M \| x∗x ∈ M+ θ } , and Mθ = 〈M+ θ 〉. When A and B are C∗-algebras, we denote by M(A) the algebra of the multipliers of A and by Mor(A,B) the set of the morphisms from A to B. 2.2. G-Products and their deformation. For the notions of an action of a l.c. group G on a C∗-algebra A, a C∗ dynamical system (A,G, α), a crossed product G α nA of A by G see [8]. The crossed product has the following universal property: For any C∗-covariant representation (π, u,B) of (A,G, α) (here B is a C∗-algebra, π : A → B a morphism, u is a group morphism from G to the unitaries of M(B), continuous for the strict topology), there is a unique morphism ρ ∈ Mor(G α n A,B) such that ρ(λt) = ut, ρ(πα(x)) = π(x) ∀t ∈ G, x ∈ A. Definition 1. Let G be a l.c. abelian group, B a C∗-algebra, λ a morphism from G to the unitary group of M(B), continuous in the strict topology of M(B), and θ a continuous action of Ĝ on B. The triplet (B, λ, θ) is called a G-product if θγ(λg) = = 〈γ, g〉λg for all γ ∈ Ĝ, g ∈ G. The unitary representation λ : G→ M(B) generates a morphism λ ∈ Mor(C∗(G), B). Identifying C∗(G) with C0(Ĝ), one gets a morphism λ ∈ Mor(C0(Ĝ), B) which is defined in a unique way by its values on the characters ug = ( γ 7→ 〈γ, g〉 ) ∈ Cb(Ĝ) : λ(ug) = λg for all g ∈ G. One can check that λ is injective. The action θ is done by: θγ(λ(ug)) = θγ(λg) = 〈γ, g〉λg = λ(ug(.− γ)). Since the ug generate Cb(Ĝ), one deduces that ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 566 P. FIMA, L. VAINERMAN θγ(λ(f)) = λ(f(.− γ)) for all f ∈ Cb(Ĝ). The following definition is equivalent to the original definition by Landstad [4] (see [5]): Definition 2. Let (B, λ, θ) be a G-poduct and x ∈ M(B). One says that x verifies the Landstad conditions if (i) θγ(x) = x for any γ ∈ Ĝ, (ii) the application g 7→ λgxλ ∗ g is continuous, (iii) λ(f)xλ(g) ∈ B for any f, g ∈ C0(Ĝ). (1) The set A ∈ M(B) verifying these conditions is a C∗-algebra called the Landstad algebra of the G-product (B, λ, θ). Definition 2 implies that if a ∈ A, then λgaλ∗g ∈ A and the map g 7→ λgaλ ∗ g is continuous. One gets then an action of G on A. One can show that the inclusion A→ M(B) is a morphism of C∗-algebras, so M(A) can be also included into M(B). If x ∈ M(B), then x ∈ M(A) if and only if(i) θγ(x) = x for all γ ∈ Ĝ, (ii) for all a ∈ A the application g 7→ λgxλ ∗ ga is continuous. (2) Let us note that two first conditions of (1) imply (2). The notions ofG-product and crossed product are closely related. Indeed, if (A,G, α) is a C∗-dynamical system with G abelian, let B = G αnA be the crossed product and λ the canonical morphism from G into the unitary group of M(B), continuous in the strict topology, and π ∈ Mor(A,B) the canonical morphism of C∗-algebras. For f ∈ K(G,A) and γ ∈ Ĝ, one defines (θγf)(t) = 〈γ, t〉f(t). One shows that θγ can be extended to the automorphisms of B in such a way that (B, Ĝ, θ) would be a C∗-dynamical system. Moreover, (B, λ, θ) is a G-product and the associated Landstad algebra is π(A). θ is called the dual action. Conversely, if (B, λ, θ) is a G-product, then one shows that there exists a C∗-dynamical system (A,G, α) such that B = G α n A. It is unique (up to a covariant isomorphism), A is the Landstad algebra of (B, λ, θ) and α is the action of G on A given by αt(x) = λtxλ ∗ t . Lemma 1 [5]. Let (B, λ, θ) be a G-product and V ⊂ A be a vector subspace of the Landstad algebra such that: λgV λ ∗ g ⊂ V for any g ∈ G, λ(C0(Ĝ))V λ(C0(Ĝ)) is dense in B. Then V is dense in A. Let (B, λ, θ) be a G-product, A its Landstad algebra, and Ψ a continuous bicharacter on Ĝ. For γ ∈ Ĝ, the function on Ĝ defined by Ψγ(ω) = Ψ(ω, γ) generates a family of unitaries λ(Ψγ) ∈ M(B). The bicharacter condition implies θγ(Uγ2) = λ(Ψγ2(.− γ1)) = Ψ(γ1, γ2)Uγ2 ∀γ1, γ2 ∈ Ĝ. One gets then a new action θΨ of Ĝ on B: θΨγ (x) = Uγθ(x)U∗γ . Note that, by commutativity of G, one has ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 A LOCALLY COMPACT QUANTUM GROUP OF TRIANGULAR MATRICES 567 θΨγ (λg) = Uγθ(λg)U∗γ = 〈γ, g〉λg ∀γ ∈ Ĝ, g ∈ G. The triplet (B, λ, θΨ) is then a G-product, called a deformed G-product. 2.3. Locally compact quantum groups [9, 10]. A pair (M,∆) is called a (von Neumann algebraic) l.c. quantum group when M is a von Neumann algebra and ∆ : M → M ⊗ M is a normal and unital ∗-homomorphism which is coassociative: (∆ ⊗ id)∆ = (id ⊗ ∆)∆ (i.e., (M,∆) is a Hopf – von Neumann algebra). There exist n.s.f. weights ϕ and ψ on M such that ϕ is left invariant in the sense that ϕ ( (ω ⊗ id)∆(x) ) = ϕ(x)ω(1) for all x ∈ M+ ϕ and ω ∈M+ ∗ , ψ is right invariant in the sense that ψ ( (id⊗ ω)∆(x) ) = ψ(x)ω(1) for all x ∈M+ ψ and ω ∈M+ ∗ . Left and right invariant weights are unique up to a positive scalar. Let us represent M on the GNS Hilbert space of ϕ and define a unitary W on H⊗H by W ∗(Λ(a)⊗ Λ(b)) = (Λ⊗ Λ)(∆(b)(a⊗ 1)) for all a, b ∈ Nφ. Here, Λ denotes the canonical GNS-map for ϕ, Λ⊗ Λ the similar map for ϕ⊗ ϕ. One proves that W satisfies the pentagonal equation: W12W13W23 = W23W12, and we say thatW is a multiplicative unitary. The von Neumann algebraM and the comultiplication on it can be given in terms of W respectively as M = { (id⊗ ω)(W ) \| ω ∈ B(H)∗ }−σ−strong∗ and ∆(x) = W ∗(1 ⊗ x)W, for all x ∈ M. Next, the l.c. quantum group (M,∆) has an antipode S, which is the unique σ-strongly* closed linear map from M to M satisfying (id⊗ω)(W ) ∈ D(S) for all ω ∈ B(H)∗ and S(id⊗ω)(W ) = (id⊗ω)(W ∗) and such that the elements (id ⊗ ω)(W ) form a σ-strong* core for S. S has a polar decomposition S = Rτ−i/2, where R (the unitary antipode) is an anti-automorphism of M and τt (the scaling group of (M,∆)) is a strongly continuous one-parameter group of automorphisms of M. We have σ(R⊗R)∆ = ∆R, so ϕR is a right invariant weight on (M,∆) and we take ψ := ϕR. Let σt be the modular automorphism group of ϕ. There exist a number ν > 0, called the scaling constant, such that ψ σt = ν−t ψ for all t ∈ R. Hence (see [11]), there is a unique positive, self-adjoint operator δM affiliated to M, such that σt(δM ) = νt δM for all t ∈ R and ψ = ϕδM . It is called the modular element of (M,∆). If δM = 1 we call (M,∆) unimodular. The scaling constant can be characterized as well by the relative invariance ϕ τt = ν−t ϕ. For the dual l.c. quantum group (M̂, ∆̂) we have M̂ = {(ω ⊗ id)(W ) \| ω ∈ B(H)∗}−σ−strong∗ and ∆̂(x) = ΣW (x⊗ 1)W ∗Σ for all x ∈ M̂. A left invariant n.s.f. weight ϕ̂ on M̂ can be constructed explicitly and the associated multiplicative unitary is Ŵ = ΣW ∗Σ. Since (M̂, ∆̂) is again a l.c. quantum group, let us denote its antipode by Ŝ, its unitary antipode by R̂ and its scaling group by τ̂t. Then we can construct the dual of ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 568 P. FIMA, L. VAINERMAN (M̂, ∆̂), starting from the left invariant weight ϕ̂. The bidual l.c. quantum group ( ˆ̂ M, ˆ̂∆) is isomorphic to (M,∆). M is commutative if and only if (M,∆) is generated by a usual l.c. group G : M = = L∞(G), (∆Gf)(g, h) = f(gh), (SGf)(g) = f(g−1), ϕG(f) = ∫ f(g)dg, where f ∈ L∞(G), g, h ∈ G and we integrate with respect to the left Haar measure dg on G. Then ψG is given by ψG(f) = ∫ f(g−1)dg and δM by the strictly positive function g 7→ δG(g)−1. L∞(G) acts on H = L2(G) by multiplication and (WGξ)(g, h) = ξ(g, g−1h), for all ξ ∈ H ⊗ H = L2(G × G). Then M̂ = L(G) is the group von Neumann algebra generated by the left translations (λg)g∈G of G and ∆̂G(λg) = λg⊗λg. Clearly, ∆̂op G := := σ ◦ ∆̂G = ∆̂G, so ∆̂G is cocommutative. (M,∆) is a Kac algebra (see [12]) if τt = id, for all t, and δM is affiliated with the center of M. In particular, this is the case when M = L∞(G) or M = L(G). We can also define the C∗-algebra of continuous functions vanishing at infinity on (M,∆) by A = [ (id⊗ ω)(W ) \| ω ∈ B(H)∗ ] and the reduced C∗-algebra (or dual C∗-algebra) of (M,∆) by Â = [ (ω ⊗ id)(W ) \| ω ∈ B(H)∗ ] . In the group case we have A = C0(G) and Â = Cr(G). Moreover, we have ∆ ∈ ∈ Mor(A,A⊗A) and ∆̂ ∈ Mor(Â, Â⊗ Â). A l.c. quantum group is called compact if ϕ(1M ) < ∞ and discrete if its dual is compact. 2.4. Twisting of locally compact quantum groups [6]. Let (M,∆) be a locally compact quantum group and Ω a unitary in M ⊗M. We say that Ω is a 2-cocycle on (M,∆) if (Ω⊗ 1)(∆⊗ id)(Ω) = (1⊗ Ω)(id⊗∆)(Ω). As an example we can consider M = L∞(G), where G is a l.c. group, with ∆G as above, and Ω = Ψ(·, ·) ∈ L∞(G×G) a usual 2-cocycle on G, i.e., a mesurable function with values in the unit circle T ⊂ C verifying Ψ(s1, s2)Ψ(s1s2, s3) = Ψ(s2, s3)Ψ(s1, s2s3) for almost all s1, s2, s3 ∈ G. This is the case for any measurable bicharacter on G. When Ω is a 2-cocycle on (M,∆), one can check that ∆Ω(·) = Ω∆(·)Ω∗ is a new coassociative comultiplication on M. If (M,∆) is discrete and Ω is any 2-cocycle on it, then (M,∆Ω) is again a l.c. quantum group (see [13], finite-dimensional case was treated in [2]). In the general case, one can proceed as follows. Let α : (L∞(G),∆G) → → (M,∆) be an inclusion of Hopf – von Neumann algebras, i.e., a faithful unital normal ∗-homomorphism such that (α⊗α)◦∆G = ∆◦α. Such an inclusion allows to construct a 2-cocycle of (M,∆) by lifting a usual 2-cocycle of G : Ω = (α ⊗ α)Ψ. It is shown in [1] that if the image of α is included into the centralizer of the left invariant weight ϕ, then ϕ is also left invariant for the new comultiplication ∆Ω. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 A LOCALLY COMPACT QUANTUM GROUP OF TRIANGULAR MATRICES 569 In particular, let G be a non commutative l.c. group and K a closed abelian subgroup of G. By Theorem 6 of [14], there exists a faithful unital normal ∗-homomorphism α̂ : L(K) → L(G) such that α̂(λKg ) = λg for all g ∈ K, and ∆̂ ◦ α̂ = (α̂⊗ α̂) ◦ ∆̂K , where λK and λ are the left regular representation of K and G respectively, and ∆̂K and ∆̂ are the comultiplications on L(K) and L(G) repectively. The composition of α̂ with the canonical isomorphism L∞(K̂) ' L(K) given by the Fourier tranformation, is a faithful unital normal *-homomorphism α : L∞(K̂) → L(G) such that ∆ ◦ α = = (α⊗α)◦∆K̂ , where ∆K̂ is the comultiplication on L∞(K̂). The left invariant weight on L(G) is the Plancherel weight for which σt(x) = δitGxδ −it G for all x ∈ L(G), where δG is the modular function of G. Thus, σt(λg) = δitG(g)λg or σt ◦ α(ug) = α(ug(· − γt)), where ug(γ) = 〈γ, g〉, g ∈ G, γ ∈ Ĝ, γt is the character K defined by 〈γt, g〉 = δ−itG (g). By linearity and density we obtain σt ◦ α(F ) = α(F (· − γt)) for all F ∈ L∞(K̂). This is why we do the following assumptions. Let (M,∆) be a l.c. quantum group, G an abelian l.c. group and α; (L∞(G),∆G) → (M,∆) an inclusion of Hopf – von Neumann algebras. Let ϕ be the left invariant weight, σt its modular group, S the antipode, R the unitary antipode, τt the scaling group. Let ψ = ϕ ◦R be the right invariant weight and σ ′ t its modular group. Also we denote by δ the modular element of (M,∆). Suppose that there exists a continuous group homomorphism t 7→ γt from R to G such that σt ◦ α(F ) = α(F (· − γt)) for all F ∈ L∞(G). Let Ψ be a continuous bicharacter on G. Notice that (t, s) 7→ Ψ(γt, γs) is a continuous bicharacter on R, so there exists λ > 0 such that Ψ(γt, γs) = λist. We define ut = λi t2 2 α(Ψ(.,−γt)) and vt = λi t2 2 α(Ψ(−γt, .)). The 2-cocycle equation implies that ut is a σt-cocyle and vt is a σ ′ t-cocycle. The Connes’ Theorem gives two n.s.f. weights on M, ϕΩ and ψΩ, such that ut = [DϕΩ : Dϕ]t and vt = [DψΩ : Dψ]t. The main result of [6] is as follows: Theorem 1. (M,∆Ω) is a l.c. quantum group with left and right invariant weight ϕΩ and ψΩ respectively. Moreover, denoting by a subscript or a superscript Ω the objects associated with (M,∆Ω) one has: τΩ t = τt, νΩ = ν and δΩ = δA−1B, D(SΩ) = D(S) and, for all x ∈ D(S), SΩ(x) = uS(x)u∗. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 570 P. FIMA, L. VAINERMAN Remark that, because Ψ is a bicharacter on G, t 7→ α(Ψ(.,−γt)) is a representation of R in the unitary group of M and there exists a positive self-adjoint operator A affiliated with M such that α(Ψ(.,−γt)) = Ait for all t ∈ R. We can also define a positive self-adjoint operator B affiliated with M such that α(Ψ(−γt, .)) = Bit. We obtain ut = λi t2 2 Ait, vt = λi t2 2 Bit. Thus, we have ϕΩ = ϕA and ψΩ = ψB , where ϕA and ψB are the weights defined by S. Vaes in [11]. One can also compute the dual C∗-algebra and the dual comultiplication. We put Lγ = α(uγ), Rγ = JLγJ for all γ ∈ Ĝ. From the representation γ 7→ Lγ we get the unital ∗-homomorphism λL : L∞(G) →M and from the representation γ 7→ Rγ we get the unital normal ∗-homomorphism λR : L∞(G) → M ′ . Let Â be the reduced C∗-algebra of (M,∆). We can define an action of Ĝ2 on Â by αγ1,γ2(x) = Lγ1Rγ2xR ∗ γ2L ∗ γ1 . Let us consider the crossed product C∗-algebra B = Ĝ2 α n Â. We will denote by λ the canonical morphism from Ĝ2 to the unitary group of M(B) continuous in the strict topology on M(B), π ∈ Mor(Â, B) the canonical morphism and θ the dual action of G2 on B. Recall that the triplet (Ĝ2, λ, θ) is a Ĝ2-product. Let us denote by (Ĝ2, λ, θΨ) the Ĝ2-product obtained by deformation of the Ĝ2-product (Ĝ2, λ, θ) by the bicharacter ω(g, h, s, t) := Ψ(g, s)Ψ(h, t) on G2. The dual deformed action θΨ is done by θΨ(g1,g2)(x) = Ug1Vg2θ(g1,g2)(x)U ∗ g1V ∗ g2 for any g1, g2 ∈ G, x ∈ B, where Ug = λL(Ψ∗g), Vg = λR(Ψg), Ψg(h) = Ψ(h, g). Considering Ψg as an element of Ĝ, we get a morphism from G to Ĝ, also noted Ψ, such that Ψ(g) = Ψg.With these notations, one has Ug = u(Ψ(−g),0) and Vg = u(0,Ψ(g)). Then the action θΨ on π(Â) is done by θΨ(g1,g2)(π(x)) = π(α(Ψ(−g1),Ψ(g2))(x)). (3) Let us consider the Landstad algebra AΨ associated with this Ĝ2-product. By definition of α and the universality of the crossed product we get a morphism ρ ∈ Mor(B,K(H)), ρ(λγ1,γ2) = Lγ1Rγ2 et ρ(π(x)) = x. (4) It is shown in [6] that ρ(AΨ) = ÂΩ and that ρ is injective on AΨ. This gives a canonical isomorphism AΨ ' ÂΩ. In the sequel we identify AΨ with ÂΩ. The comultiplication ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 A LOCALLY COMPACT QUANTUM GROUP OF TRIANGULAR MATRICES 571 can be described in the following way. First, one can show that, using universality of the crossed product, there exists a unique morphism Γ ∈ Mor(B,B ⊗B) such that Γ ◦ π = (π ⊗ π) ◦ ∆̂ and Γ(λγ1,γ2) = λγ1,0 ⊗ λ0,γ2 . Then we introduce the unitary Υ = (λR ⊗ λL)(Ψ̃) ∈ M(B ⊗ B), where Ψ̃(g, h) = = Ψ(g, gh). This allows us to define the ∗-morphism ΓΩ(x) = ΥΓ(x)Υ∗ from B to M(B⊗B). One can show that ΓΩ ∈ Mor(AΨ, AΨ⊗AΨ) is the comultiplication on AΨ. Note that if M = L(G) and K is an abelian closed subgroup of G, the action α of K2 on C0(G) is the left-right action. 3. Twisting of the group of 2×2 upper triangular matrices with determinant 1. Consider the following subgroup of SL2(C) : G := {( z ω 0 z−1 ) , z ∈ C∗, ω ∈ C } . Let K ⊂ G be the subgroup of diagonal matrices in G, i.e.., K = C∗. The elements of G will be denoted by (z, ω), z ∈ C, ω ∈ C∗. The modular function of G is δG((z, ω)) = \|z\|−2. Thus, the morphism (t 7→ γt) from R to Ĉ∗ is given by 〈γt, z〉 = \|z\|2it for all z ∈ C∗, t ∈ R. We can identify Ĉ∗ with Z× R∗+ in the following way: Z× R∗+ → Ĉ∗, (n, ρ) 7→ γn,ρ = (reiθ 7→ ei ln r ln ρeinθ). Under this identification, γt is the element (0, et) of Z× R∗+. For all x ∈ R, we define a bicharacter on Z× R∗+ by Ψx((n, ρ), (k, r)) = eix(k ln ρ−n ln r). We denote by (Mx,∆x) the twisted l.c. quantum group. We have Ψx((n, ρ), γ−1 t ) = eixtn = ueixt((n, ρ)). In this way we obtain the operator Ax deforming the Plancherel weight Aitx = α(ueixt) = λG(eitx,0). In the same way we compute the operator Bx deforming the Plancherel weight Bitx = λG(e−ixt,0) = A−itx . Thus, we obtain for the modular element δitx = A−itx Bitx = λG(e−2itx,0). The antipode is not deformed. The scaling group is trivial but, if x 6= 0, (Mx,∆x) is not a Kac algebra because δx is not affiliated with the center of M. Let us look if (Mx,∆x) can be isomorphic for different values of x. One can remark that, since Ψ−x = Ψ∗x ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 572 P. FIMA, L. VAINERMAN is antisymmetric and ∆ is cocommutative, we have ∆−x = σ∆x, where σ is the flip on L(G) ⊗ L(G). Thus, (M−x,∆−x) ' (Mx,∆x)op, where ”op” means the opposite quantum group. So, it suffices to treat only strictly positive values of x. The twisting deforms only the comultiplication, the weights and the modular element. The simplest invariant distinguishing the (Mx,∆x) is then the specter of the modular element. Using the Fourier transformation in the first variable, on has immediately Sp(δx) = qZ x ∪ {0}, where qx = e−2x. Thus, if x 6= y, x > 0, y > 0, one has qZ x 6= qZ y and, consequently, (Mx,∆x) and (My,∆y) are non isomorphic. We compute now the dual C∗-algebra. The action of K2 on C0(G) can be lifted to its Lie algebra C2. The lifting does not change the result of the deformation (see [5], Proposition 3.17) but simplify calculations. The action of C2 on C0(G) will be denoted by ρ. One has ρz1,z2(f)(z, ω) = f(ez2−z1z, e−(z1+z2)ω). (5) The group C is self-dual, the duality is given by (z1, z2) 7→ exp (iIm(z1z2)) . The generators uz, z ∈ C, of C0(C) are given by uz(w) = exp (iIm(zw)) , z, w ∈ C. Let x ∈ R. We will consider the following bicharacter on C : Ψx(z1, z2) = exp ( ixIm(z1z2) ) . Let B be the crossed product C∗-algebra C2 nC0(G). We denote by ((z1, z2) 7→ λz1,z2) the canonical group homomorphism from G to the unitary group of M(B), continuous for the strict topology, and π ∈ Mor(C0(G), B) the canonical homomorphism. Also we denote by λ ∈ Mor(C0(G2), B) the morphism given by the representation ((z1, z2) 7→ 7→ λz1,z2). Let θ be the dual action of C2 on B. We have, for all z, w ∈ C, Ψx(w, z) = = uxz(w). The deformed dual action is given by θΨx z1,z2(b) = λ−xz1,xz2θz1,z2(b)λ ∗ −xz1,xz2 . (6) Recall that θΨx z1,z2(λ(f)) = θz1,z2(λ(f)) = λ(f(· − z1, · − z2)) ∀f ∈ Cb(C2). (7) Let Âx be the associated Landstad algebra. We identify Âx with the reduced C∗-algebra of (Mx,∆x). We will now construct two normal operators affiliated with Âx, which generate Âx. Let a and b be the coordinate functions on G, and α = π(a), β = π(b). Then α and β are normal operators, affiliated with B, and one can see, using (5), that λz1,z2αλ ∗ z1,z2 = ez2−z1α, λz1,z2βλ ∗ z1,z2 = e−(z1+z2)β. (8) We can deduce, using (6), that ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 A LOCALLY COMPACT QUANTUM GROUP OF TRIANGULAR MATRICES 573 θΨx z1,z2(α) = ex(z1+z2)α, θΨx z1,z2(β) = ex(z1−z2)β. (9) Let Tl and Tr be the infinitesimal generators of the left and right shift respectively, i.e., Tl and Tr are normal, affiliated with B, and λz1,z2 = exp ( i Im(z1Tl) ) exp ( i Im(z2Tr) ) for all z1, z2 ∈ C. Thus, we have λ(f) = f(Tl, Tr) for all f ∈ Cb(C2). Let U = λ(Ψx), we define the following normal operators affiliated with B : α̂ = U∗αU, β̂ = UβU∗. Proposition 1. The operators α̂ and β̂ are affiliated with Âx and generate Âx. Proof. First let us show that f(α̂), f(β̂) ∈ M(Ât) for all f ∈ C0(C). One has, using (7): θΨx z1,z2(U) = λ ( Ψx(.− z1, .− z2) ) = = UeixIm(−z2Tl) eixIm(z1Tr)Ψx(z1, z2) = Uλ−xz2,xz1Ψx(z1, z2). Now, using (9) and (8), we obtain θΨx z1,z2(α̂) = α̂, θΨx z1,z2(β̂) = β̂ for all z1, z2 ∈ C. Thus, for all f ∈ C0(C), f(α̂) and f(β̂) are fixed points for the action θΨx . Let f ∈ C0(C). Using (8) we find λz1,z2f(α̂)λ∗z1,z2 = U∗f(ez2−z1α)U, λz1,z2f(β̂)λ∗z1,z2 = U∗f(e−(z1+z2)β)U. (10) Because f is continuous and vanish at infinity, the applications (z1, z2) 7→ λz1,z2f(α̂)λ∗z1,z2 and (z1, z2) 7→ λz1,z2f(β̂)λ∗z1,z2 are norm-continuous and f(α̂), f(β̂) ∈M(Âx) for all f ∈ C0(C). Taking in mind Proposition 4 (see Appendix), in order to show that α̂ is affiliated with Âx, it suffices to show that the vector space I generated by f(α̂)a, with f ∈ C0(C) and a ∈ Âx, is dense in Âx. Using (10), we see that I is globally invariant under the action implemented by λ. Let g(z) = (1 + zz)−1. As λ(C0(C2))U = λ(C0(C2)), we can deduce that the closure of λ ( C0(C2) ) g(α̂)Âxλ ( C0(C2) ) is equal to[ λ(C0(C2))(1 + α∗α)−1U∗Âxλ(C0(C2)) ] . As the set U∗Âxλ ( C0(C2) ) is dense in B and α is affiliated with B, the set λ ( C0(C2) ) (1 + α∗α)−1U∗Âxλ ( C0(C2) ) is dense in B. Moreover, it is included in λ ( C0(C2) ) Iλ ( C0(C2) ) , so λ ( C0(C2) ) Iλ ( C0(C2) ) is dense in B. We conclude, using Lemma 1, that I is dense in Âx. One can show in the same way that β̂ is affiliated with Âx. ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 574 P. FIMA, L. VAINERMAN Now, let us show that α̂ and β̂ generate Âx. By Proposition 5, it suffices to show that V = 〈 f(α̂)g(β̂), f, g ∈ C0(C) 〉 is a dense vector subspace of Âx. We have shown above that the elements of V satisfy the two first Landstad’s conditions. Let W = [ λ ( C0(C2) ) Vλ(C0(C2)) ] . We will show thatW = B. This proves that the elements of V satisfy the third Landstad’s condition, and then V ⊂ Âx. Then (10) shows that V is globally invariant under the action implemented by λ, so V is dense in Âx by Lemma 1. One has: W = [ xU∗f(α)U2g(β)U∗y, f, g ∈ C0(C), x, y ∈ λ(C0(C2)) ] . Because U is unitary, we can substitute x with xU and y with Uy without changing W : W = [ xf(α)U2g(β)y, f, g ∈ C0(C), x, y ∈ λ ( C0(C2) )] . Using, for all f ∈ C0(C), the norm-continuity of the application (z1, z2) 7→ λz1,z2f(α)λ∗z1,z2 = ez2−z1α, one deduces that[ f(α)x, f ∈ C0(C), x ∈ λ(C0(C2)) ] = [ xf(α), f ∈ C0(C), x ∈ λ(C0(C2)) ] . In particular, W = [ f(α)xU2g(β)y, f, g ∈ C0(C), x, y ∈ λ(C0 ( C2) )] . Now we can commute g(β) and y, and we obtain W = [ f(α)xU2yg(β), f, g ∈ C0(C), x, y ∈ λ(C0 ( C2) )] . Substituting x 7→ xU∗, y 7→ U∗y, one has W = [ f(α)xyg(β), f, g ∈ C0(C), x, y ∈ λ(C0 ( C2) )] . Commuting back f(α) with x and g(β) with y, we obtain W = [ xf(α)g(β)y, f, g ∈ C0(C), x, y ∈ λ(C0(C2)) ] = B. This concludes the proof. We will now find the commutation relations between α̂ and β̂. Proposition 2. One has: 1) α et T ∗l + T ∗r strongly commute and α̂ = ex(T ∗ l +T∗r )α; 2) β et T ∗l − T ∗r strongly commute and β̂ = ex(T ∗ l −T ∗ r )β. Thus, the polar decompositions are given by Ph(α̂) = e−ixIm(Tl+Tr)Ph(α), \|α̂\| = exRe(Tl+Tr)\|α\|, ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 A LOCALLY COMPACT QUANTUM GROUP OF TRIANGULAR MATRICES 575 Ph(β̂) = e−ixIm(Tl−Tr)Ph(β), \|β̂\| = exRe(Tl−Tr)\|β\|. Moreover, we have the following relations: 1) \|α̂\| and \|β̂\| strongly commute, 2) Ph(α̂)Ph(β̂) = Ph(β̂)Ph(α̂), 3) Ph(α̂)\|β̂\|Ph(α̂)∗ = e4x\|β̂\|, 4) Ph(β̂)\|α̂\|Ph(β̂)∗ = e4x\|α̂\|. Proof. Using (8), we find, for all z ∈ C : eiIm(z(T∗l +T∗r ))αe−iIm(z(T∗l +T∗r )) = λ−z,−zαλ ∗ −z,−z = e−z+zα = α. Thus, T ∗l + T ∗r and α strongly commute. Moreover, because eixImTlT ∗ l = 1, one has α̂ = e−ixImTlT ∗ r αeixImTlT ∗ r = e−ixImTl(Tl+Tr)∗αeixImTl(Tl+Tr)∗ . We can now prove the point 1 using the equality e−ixImTlωαeixImTlω = exωα, the preceding equation and the fact that T ∗l +T ∗r and α strongly commute. The proof of the second assertion is similar and the polar decompositions follows. From (8) we deduce e−ixIm(Tr−Tl)αeixIm(Tr−Tl) = e−2xα, eixIm(Tl+Tr)βe−ixIm(Tl+Tr) = e−2xβ, eixRe(Tr−Tl)αe−ixRe(Tr−Tl) = e2ixα, eixRe(Tl+Tr)βe−ixRe(Tl+Tr) = e−2ixβ. It is now easy to prove the last relations from the preceding equations and the polar decompositions. The proposition is proved. We can now give a formula for the comultiplication. Proposition 3. Let ∆̂x be the comultiplication on Âx. One has ∆̂x(α̂) = α̂⊗ α̂, ∆̂x(β̂) = α̂⊗ β̂+̇β̂ ⊗ α̂−1. Proof. Using the Preliminaries, we have that ∆̂x = ΥΓ(.)Υ∗, where Υ = eixImTr⊗T∗l and Γ is given by Γ(Tl) = Tl ⊗ 1, Γ(Tr) = 1⊗ Tr; Γ restricted to C0(G) is equal to the comultiplication ∆G. Define R = ΥΓ(U∗). One has ∆x(α̂) = R(α⊗α)R∗. Thus, it is sufficient to show that (U ⊗ U)R commute with α⊗ α. Indeed, in this case, one has ∆̂x(α̂) = R(α⊗α)R∗ = (U∗⊗U∗)(U ⊗U)R(α⊗α)R∗(U∗⊗U∗)(U ⊗U) = α̂⊗ α̂. Let us show that (U ⊗U)R commute with α⊗α. From the equality U = eixImTlT ∗ r , we deduce that Γ(U∗) = e−ixImTl⊗T∗r , U ⊗ U = eixIm(TlT ∗ r ⊗1+1⊗TlT ∗ r ). Thus, R = e−ixIm(T∗r ⊗Tl+Tl⊗T∗r ) and ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 576 P. FIMA, L. VAINERMAN (U ⊗ U)R = eixIm(TlT ∗ r ⊗1+1⊗TlT ∗ r −T ∗ r ⊗Tl−Tl⊗T∗r ). Notice that TlT ∗ r ⊗ 1 + 1⊗ TlT ∗ r − T ∗r ⊗ Tl − Tl ⊗ T ∗r = (Tl ⊗ 1− 1⊗ Tl)(T ∗r ⊗ 1− 1⊗ T ∗r ). Thus, it suffices to show that Tl ⊗ 1 − 1 ⊗ Tl and T ∗r ⊗ 1 − 1 ⊗ T ∗r strongly commute with α⊗ α. This follows from the equations eiImz(T ∗ r ⊗1−1⊗T∗r )(α⊗ α)e−iImz(T ∗ r ⊗1−1⊗T∗r ) = = (λ0,−z ⊗ λ0,z)(α⊗ α)(λ0,−z ⊗ λ0,z)∗ = e−zezα⊗ α = α⊗ α ∀z ∈ C and eiImz(Tl⊗1−1⊗Tl)(α⊗ α)e−iImz(Tl⊗1−1⊗Tl) = = (λz,0 ⊗ λ−z,0)(α⊗ α)(λz,0 ⊗ λ−z,0)∗ = = e−zezα⊗ α = α⊗ α ∀z ∈ C. Put S = ΥΓ(U). One has ∆̂x(β̂) = S(α⊗ β + β ⊗ α−1)S∗ = S(α⊗ β)S∗+̇S(β ⊗ α−1)S∗. As before, we see that it suffices to show that (U ⊗ U∗)S commutes with α ⊗ β and that (U∗ ⊗ U)S commutes with β ⊗ α−1, and one can check this in the same way. The proposition is proved. Let us summarize the preceding results in the following corollary (see [15, 5] for the definition of commutation relation between unbounded operators): Corollary 1. Let q = e8x. The C∗-algebra Âx is generated by 2 normal operators α̂ and β̂ affiliated with Âx such that α̂β̂ = β̂α̂, α̂β̂∗ = qβ̂∗α̂. Moreover, the comultiplication ∆̂x is given by ∆̂x(α̂) = α̂⊗ α̂, ∆̂x(β̂) = α̂⊗ β̂+̇β̂ ⊗ α̂−1. Remark 1. One can show, using the results of [6], that the application (q 7→ Wq) which maps the parameter q to the multiplicative unitary of the twisted l.c. quantum group is continuous in the σ-weak topology. 4. Appendix. Let us cite some results on operators affiliated with a C∗-algebra. Proposition 4. Let A ⊂ B(H) be a non degenerated C∗-subalgebra and T a normal densely defined closed operator on H. Let I be the vector space generated by f(T )a, where f ∈ C0(C) and a ∈ A. Then (TηA) ⇔ ( cf(T ) ∈ M(A) for any f ∈ C0(C) et I is dense in A ) . Proof. If T is affiliated withA, then it is clear that f(T ) ∈ M(A) for any f ∈ C0(C), and that I is dense in A (because I contains (1 + T ∗T )− 1 2A). To show the converse, ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 consider the ∗-homomorphism πT : C0(C) → M(A) given by πT (f) = f(T ). By hypothesis, πT ( C0(C) ) A is dense in A. So, πT ∈ Mor(C0(C), A) and T = πT (z 7→ z) is then affiliated with A. Proposition 5. LetA ⊂ B(H) be a non degeneratedC∗-subalgebra and T1, T2, . . . . . . , TN normal operators affiliated withA. Let us denote by V the vector space generated by the products of the form f1(T1) f2(T2) . . . fN (TN ), with fi ∈ C0(C). If V is a dense vector subspace of A, then A is generated by T1, T2, . . . , TN . Proof. This follows from Theorem 3.3 in [16]. 1. Enock M., Vainerman L. Deformation of a Kac algebra by an abelian subgroup // Communs Math. Phys. – 1996. – 178, № 3. – P. 571 – 596. 2. Vainerman L. 2-Cocycles and twisting of Kac algebras // Ibid. – 1998. – 191, № 3. – P. 697 – 721. 3. Rieffel M. Deformation quantization for actions of Rd // Mem. AMS. – 1993. – 506. 4. Landstad M. Quantization arising from abelian subgroups // Int. J. Math. – 1994. – 5. – P. 897 – 936. 5. Kasprzak P. Deformation of C∗-algebras by an action of abelian groups with dual 2-cocycle and quantum groups. – Preprint: arXiv:math.OA/0606333. 6. Fima P., Vainerman L. Twisting of locally compact quantum groups. Deformation of the Haar measure. – Preprint. – 2000. 7. Stratila S. Modular theory in operator algebras. – Turnbridge Wells, England: Abacus Press, 1981. 8. Pedersen G. K. C∗-algebras and their automorphism groups. – Acad. Press, 1979. 9. Kustermans J., Vaes S. Locally compact quantum groups // Ann. sci. Ecole norm. super. Ser. 33. – 2000. – 4, № 6. – P. 547 – 934. 10. Kustermans J., Vaes S. Locally compact quantum groups in the von Neumann algebraic setting // Math. scand. – 2003. – 92 № 1. – P. 68 – 92. 11. Vaes S. A Radon – Nikodym theorem for von Neuman algebras // J. Oper. Theory. – 2001. – 46, № 3. – P. 477 – 489. 12. Enock M., Schwartz J.-M. Kac algebras and duality of locally compact groups. – Berlin: Springer, 1992. 13. Bichon J., De Rijdt J. A., Vaes S. Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups // Communs Math. Phys. – 2006. – 22. – P. 703 – 728. 14. Takesaki M., Tatsuuma N. Duality and subgroups // Ann. Math. – 1971. – 93. – P. 344 – 364. 15. Woronowicz S. L. Quantum E(2) group and its Pontryagin dual // Lett. Math. Phys. – 1991. – 23. – P. 251 – 263. 16. Woronowicz S. L. C∗-algebras generated by unbounded elements // Revs Math. Phys. – 1995. – 7, № 3. – P. 481 – 521. Received 30.10.07

A locally compact quantum group of triangular matrices

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