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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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 |
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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. |
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Article |
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Fima, P. Vainerman, L. |
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Fima, P. Vainerman, L. |
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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 |
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A locally compact quantum group of triangular matrices |
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A locally compact quantum group of triangular matrices |
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locally compact quantum group of triangular matrices |
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Інститут математики НАН України |
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2008 |
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Статті |
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https://nasplib.isofts.kiev.ua/handle/123456789/164494 |
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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. |
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Український математичний журнал |
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AT fimap alocallycompactquantumgroupoftriangularmatrices AT vainermanl alocallycompactquantumgroupoftriangularmatrices AT fimap lokalʹnokompaktnakvantovagrupatrikutnihmatricʹ AT vainermanl lokalʹnokompaktnakvantovagrupatrikutnihmatricʹ AT fimap locallycompactquantumgroupoftriangularmatrices AT vainermanl locallycompactquantumgroupoftriangularmatrices |
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2025-12-01T20:07:07Z |
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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
 =
[
(ω ⊗ id)(W ) | ω ∈ B(H)∗
]
.
In the group case we have A = C0(G) and  = 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∗.
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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  be the reduced C∗-algebra of (M,∆). We can define an action
of Ĝ2 on  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
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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
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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
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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.
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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|,
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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
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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,
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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].
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Received 30.10.07
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