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TWISTING OF THE QUANTUM DOUBLE AND THE
WEYL ALGEBRA
BYUNG-JAY KAHNG
Abstract. Quantum double construction, originally due to Drinfeld
and has been since generalized even to the operator algebra framework,
is naturally associated with a certain (quasitriangular) R-matrix R. It
turns out that R determines a twisting of the comultiplication on the
quantum double. It then suggests a twisting of the algebra structure on
the dual of the quantum double. For D(G), the C
∗ -algebraic quantum
double of an ordinary group G, the “twisted
̂ D(G)” turns out to be the
Weyl algebra C 0 (G) ×τ G, which is in turn isomorphic to K(L
2 (G)).
This is the C
∗ -algebraic counterpart to an earlier (finite-dimensional)
result by Lu. It is not so easy technically to extend this program to
the general locally compact quantum group case, but we propose here
some possible approaches, using the notion of the (generalized) Fourier
transform.
- Introduction
There are a few different approaches to formulate the notion of quan-
tum groups, which are generalizations of ordinary groups. In the finite-
dimensional case, they usually come down to Hopf algebras [1], [14], although
there actually exist examples of quantum groups that cannot be described
only by Hopf algebra languages. More generally, the approaches to quan-
tum groups include the (purely algebraic) setting of “quantized universal
enveloping (QUE) algebras” [6], [4]; the setting of multiplier Hopf algebras
and algebraic quantum groups [19], [9]; and the (C
∗
gebraic) setting of locally compact quantum groups [10], [11], [13], [20]. In
this paper, we are mostly concerned with the setting of C
∗ -algebraic locally
compact quantum groups.
In all these approaches to quantum groups, one important aspect is that
the category of quantum groups is a “self-dual” category, which is not the
case for the (smaller) category of ordinary groups. To be more specific, a
typical quantum group A is associated with a certain dual object
A, which
is also a quantum group, and the dual object,
A, of the dual quantum group
is actually isomorphic to A. This result,
A
= A, is a generalization of the
Pontryagin duality, which holds in the smaller category of abelian locally
compact groups.
For a finite dimensional Hopf algebra H, its dual object is none other than
the dual vector space H
′ , with its Hopf algebra structure obtained naturally
1
2 BYUNG-JAY KAHNG
from that of H. In general, however, a typical quantum group A would be
infinite dimensional, and in that case, the dual vector space is too big to be
given any reasonable structure (For instance, one of the many drawbacks is
that (A ⊗ A)
′ is strictly larger than A
′ ⊗ A
′ .).
In each of the approaches to quantum groups, therefore, a careful at-
tention should be given to making sense of what the dual object is for a
quantum group, as well as to exploring the relationship between them. This
is especially true for the analytical settings, where the quantum groups are
required to have additional, topological structure. The success of the lo-
cally compact quantum group framework by Kustermans and Vaes [10], and
also by Masuda, Nakagami, and Woronowicz [13] is that they achieve the
definition of locally compact quantum groups so that it has the self-dual
property.
Meanwhile, given a Hopf algebra H and its dual
H, there exists the notion
of the “quantum double” HD =
H
op on H (see [6], [14]). This notion can be
generalized even to the setting of locally compact quantum groups: From
a von Neumann algebraic quantum group (N, ∆), one can construct the
quantum double (N D
D
). See Section 2 below.
The quantum double is associated with a certain “quantum universal R-
matrix” type operator R ∈ N D
⊗ N
D
. It turns out that R determines a left
cocycle for ∆ D
, and allows us to twist (or deform) the comultiplication on
N
D
, or its C
∗ -algebraic counterpart A D
. The result, (A D
, R∆
D
), can no
longer become a locally compact quantum group, but it suggests a twisting
of the algebra structure at the level of
A
D
, the dual of the quantum double.
Our intention here is to explore this algebra, the “deformed
A
D
There are two crucial obstacles in carrying out this program. For one
thing, the C
∗ -algebra
A
D
itself can be rather complicated in general. In
addition, unlike in the algebraic approaches, even the simple tool like the
dual pairing is not quite easy to work with. In the locally compact quantum
group framework, the dual pairing between a quantum group A and its dual
A is defined at dense subalgebra level, by using the multiplicative unitary
operator associated with A and
A. While it is a correct definition (in the
sense that it is a natural generalization of the obvious dual pairing between
H and H
′ in the finite-dimensional case), the way it is defined makes it
rather difficult to work with. For instance, there is no straightforward way
of obtaining a dual object of a C
∗ -bialgebra.
These technical difficulties cannot be totally overcome, but we can im-
prove the situation by having a better understanding of the duality picture.
Recently in [8], motivated by Van Daele’s work in the multiplier Hopf alge-
bra framework [21], the author defined the (generalized) Fourier transform
between a locally compact quantum group and its dual. In addition, an
alternative description of the dual pairing was found (see Section 4 of [8]),
in terms of the Haar weights and the Fourier transform. This alternative
perspective to the dual pairing is useful in our paper.
4 BYUNG-JAY KAHNG
will collect some information that can be used in our efforts to go further into
the case of general locally compact quantum groups. We will propose here a
reasonable description for the deformed
A
D
. The notion of the generalized
Fourier transform defined in [8] will play a central role.
- Preliminaries
2.1. Locally compact quantum groups. Let us first begin with the defi-
nition of a von Neumann algebraic locally compact quantum group, given by
Kustermans and Vaes [11]. This definition is known to be equivalent to the
definition in the C
∗ -algebra setting [10], and also to the formulation given
by Masuda–Nakagami–Woronowicz [13]. Refer also to the recent paper by
Van Daele [20]. We note that the existence of the Haar (invariant) weights
has to be assumed as a part of the definition.
Definition 2.1. Let M be a von Neumann algebra, together with a unital
normal
∗ -homomorphism ∆ : M → M ⊗ M satisfying the “coassociativity”
condition: (∆ ⊗ id)∆ = (id ⊗∆)∆. Assume further the existence of a left
invariant weight and a right invariant weight, as follows:
- ϕ is an n.s.f. weight on M that is left invariant:
ϕ
(ω ⊗ id)(∆x)
= ω(1)ϕ(x), for all x ∈ M
ϕ
and ω ∈ M
∗
- ψ is an n.s.f. weight on M that is right invariant:
ψ
(id ⊗ω)(∆x)
= ω(1)ψ(x), for all x ∈ M
ψ
and ω ∈ M
∗
Then we say that (M, ∆) is a von Neumann algebraic quantum group.
Remark. We are using the standard notations and terminologies from the
theory of weights. For instance, an “n.s.f. weight” is a normal, semi-finite,
faithful weight. For an n.s.f. weight ϕ, we write x ∈ M
ϕ
to mean x ∈ M
so that ϕ(x) < ∞, while x ∈ N ϕ
means x ∈ M so that ϕ(x
∗ x) < ∞. See
[17]. Meanwhile, it can be shown that the Haar weights ϕ and ψ above are
unique, up to scalar multiplication.
Let us fix ϕ. Then by means of the GNS construction (H, ι, Λ) for ϕ,
we may as well regard M as a subalgebra of the operator algebra B(H),
such as M = ι(M ) ⊆ B(H). Thus we will have:
Λ(x), Λ(y)
= ϕ(y
∗ x) for
x, y ∈ Nϕ, and aΛ(y) = ι(a)Λ(y) = Λ(ay) for y ∈ Nϕ, a ∈ M. Consider
next the operator T , which is the closure of the map Λ(x) 7 → Λ(x
∗ ) for
x ∈ N ϕ
∩ N
∗
ϕ
. Expressing its polar decomposition as T = J∇
1 / 2 , we obtain
in this way the “modular operator” ∇ and the “modular conjugation” J.
The operator ∇ determines the modular automorphism group. Refer to the
standard weight theory [17].
Meanwhile, there exists a unitary operator W ∈ B(H⊗H), called the mul-
tiplicative unitary operator for (M, ∆). It is defined by W
∗
Λ(x) ⊗ Λ(y)
(∆y)(x ⊗ 1)
, for x, y ∈ N ϕ
. It satisfies the pentagon equa-
tion of Baaj and Skandalis [2]: W 12
W
13
W
23
= W
23
W
12
. We also have:
QUANTUM DOUBLE AND THE WEYL ALGEBRA 5
∆a = W
∗ (1 ⊗ a)W , for a ∈ M. The operator W is the “left regular repre-
sentation”, and it provides the following useful characterization of M :
M = {(id ⊗ω)(W ) : ω ∈ B(H)∗}
w
⊆ B(H)
where −
w denotes the von Neumann algebra closure (for instance, the closure
under σ-weak topology).
If we wish to consider the quantum group in the C
∗ -algebra setting, we
just need to take the norm completion instead, and restrict ∆ to A. See
[10], [20]. Namely,
A = {(id ⊗ω)(W ) : ω ∈ B(H) ∗
⊆ B(H)
Constructing the antipode is rather technical (it uses the right Haar
weight), and we refer the reader to the main papers [10], [11]. See also
an improved treatment given in [20], where the antipode is defined in a
more natural way by means of Tomita–Takesaki theory. For our purposes,
we will just mention the following useful characterization of the antipode S:
S
(id ⊗ω)(W )
= (id ⊗ω)(W
∗ ). (2.1)
In fact, the subspace consisting of the elements (id ⊗ω)(W ), for ω ∈ B(H) ∗
is dense in M and forms a core for S. Meanwhile, there exist a unique
∗ -antiautomorphism R (called the “unitary antipode”) and a unique contin-
uous one parameter group τ on M (called the “scaling group”) such that we
have: S = Rτ −
i
2
. Since (R ⊗ R)∆ = ∆
cop R, where ∆
cop is the co-opposite
comultiplication (i. e. ∆
cop = χ ◦ ∆, for χ the flip map on M ⊗ M ), the
weight ϕ ◦ R is right invariant. So we can, without loss of generality, choose
ψ to equal ϕ ◦ R. The GNS map for ψ will be written as Γ.
From the right Haar weight ψ, we can find another multiplicative unitary
V , defined by V
Γ(x) ⊗ Γ(y)
∆x)(1 ⊗ y)
, for x, y ∈ N ψ
. It is
the “right regular representation”, and it provides an alternative character-
ization of M : That is, M = {(ω ⊗ id)(V ) : ω ∈ B(H) ∗
w
⊆ B(H)
Next, let us consider the dual quantum group. Working with the other
leg of the multiplicative unitary operator W , we define:
M =
(ω ⊗ id)(W ) : ω ∈ B(H)∗
}w (
⊆ B(H)
This is indeed shown to be a von Neumann algebra. We can define a comul-
tiplication on it, by
∆(y) = ΣW (y ⊗ 1)W
∗ Σ, for all y ∈
M. Here, Σ is the
flip map on H ⊗ H, and defining the dual comultiplication in this way makes
it “flipped”, unlike in the purely algebraic settings (See the remark following
Proposition 2.2 for more discussion.). This is done for technical reasons, so
that it is simpler to work with the multiplicative unitary operator.
The general theory assures that (
M ,
∆) is again a von Neumann algebraic
quantum group, together with appropriate Haar weights ˆϕ and
ψ. By tak-
ing the norm completion, we can consider the C
∗ -algebraic quantum group
A,
∆). The operator
W = ΣW
∗ Σ is easily seen to be the multiplicative
unitary for (
M ,
∆). It turns out that W ∈ M ⊗
M and
W ∈
M ⊗ M.
QUANTUM DOUBLE AND THE WEYL ALGEBRA 7
This definition is suggested by [2]. The properties of this pairing map is
given below:
Proposition 2.2. Let (M, ∆) and (
M ,
∆) be the dual pair of locally compact
quantum groups, and let A and
A be their dense subalgebras, as defined
above. Then the map 〈 | 〉 :
A × A → C, given by equation (2.3), is a valid
dual pairing. Moreover, we have:
(1) 〈b 1 b 2 | a〉 =
b 1 ⊗ b 2 | ∆(a)
, for a ∈ A, b 1 , b 2 ∈
A.
(2) 〈b | a 1
a 2
cop (b) | a 1
⊗ a 2
〉, for a 1
, a 2
∈ A, b ∈
A.
b | S(a)
S
− 1 (b) | a
, for a ∈ A, b ∈
A.
Remark. Bilinearity of 〈 | 〉 is obvious, and the proof of the three properties
is straightforward. See, for instance, Proposition 4.2 of [8]. Except for the
appearance of the co-opposite comultiplication
cop in (2), the proposition
shows that 〈 | 〉 is a suitable dual pairing map that generalizes the pairing
map on (finite-dimensional) Hopf algebras. The difference is that in purely
algebraic frameworks (Hopf algebras, QUE algebras, or even multiplier Hopf
algebras), the dual comultiplication on H
′ is simply defined by dualizing the
product on H via the natural pairing map between H and H
′
. Whereas in
our case, the pairing is best defined using the multiplicative unitary operator.
It turns out that defining as we have done the dual comultiplication as
“flipped” makes things to become technically simpler, even with (2) causing
minor annoyance.
Meanwhile, let us quote below an alternative description given in [8] of this
pairing map, using the Haar weights and the generalized Fourier transform.
The new descriptions are only valid on certain subspaces D ⊆ A and
D ⊆
A,
but D and
D are dense subalgebras in M and
M respectively, and form cores
for the antipode maps S and
S.
Theorem 2.3. Let D ⊆ A and
D ⊆
A be the dense subalgebras as defined
in Section 4 of [8]. Then:
(1) For a ∈ D, its Fourier transform is defined by
F(a) := (ϕ ⊗ id)
W (a ⊗ 1)
(2) For b ∈
D, the inverse Fourier transform is defined by
F
− 1
(b) := (id ⊗ ϕˆ)
W
∗
(1 ⊗ b)
(3) The dual pairing map 〈 | 〉 :
A × A → C given in Proposition 2.
takes the following form, if we restrict it to the level of D and
D:
〈b | a〉 =
Λ(b), Λ(a
∗
)
= ϕ
aF
− 1
(b)
= ˆϕ
F(a
∗
)
∗
b
= (ϕ ⊗ ϕˆ)
[
(a ⊗ 1)W
∗ (1 ⊗ b)
]
Remark. Here, ϕ and ˆϕ are the left invariant Haar weights for (M, ∆) and
M ,
∆), while Λ and
Λ are the associated GNS maps. The maps F and F
− 1
are actually defined in larger subspaces, but we restricted the domains here
8 BYUNG-JAY KAHNG
to D and
D, for convenience. As in the classical case, the Fourier inversion
theorem holds:
F
− 1
F(a)
= a, a ∈ D, and F
F
− 1 (b)
= b, b ∈
D.
See [8] for more careful discussion on all these, including the definition of
the Fourier transform and the proof of the result on the dual pairing.
- The quantum double
The quantum double construction was originally introduced by Drinfeld
[6], in the Hopf algebra framework. The notion can be extended to the
setting of locally compact quantum groups. See [23] (also see [7], and some
earlier results in [15] and Section 8 of [2]). Some different formulations exist,
but all of them are special cases of a more generalized notion of a double
crossed product construction developed recently by Baaj and Vaes [3]. While
we do not plan to go to the full generality as in that paper, let us give here
the definition adapted from [3].
Let (N, ∆N ) be a locally compact quantum group, and let WN be its mul-
tiplicative unitary operator. Write (M 1
1
) = (N, ∆
cop
N
) and (M 2
2
N ,
N
). Suggested by Proposition 8.1 of [3], consider the operators K and
K on H ⊗ H:
K = W
N
J 1 ⊗ J 2 )W
∗
N
K = W
N
(J 1 ⊗
J 2 )W
∗
N
where J 1
J
1
, J
2
J
2
are the modular conjugations for M 1
M
1
, M
2
M
2
. In
our case, we would actually have:
J
1
= J
2
and
J
2
= J
1
. Next, following
Notation 3.2 of [3], write:
Z = K
K(
J
1
J
1
J
2
J
2
Then on H ⊗ H ⊗ H ⊗ H, define the unitary operator:
W
m
= (ΣV
∗
1
13
Z
∗
34
W
2 , 24
Z
34
where V 1
(right regular representation of M 1
) and W 2
(left regular represen-
tation of M 2
) are multiplicative unitary operators associated with M 1
and
M
2
. By Proposition 3.5 and Theorem 5.3 of [3], the operator W m
is a mul-
tiplicative unitary operator, and it gives rise to a locally compact quantum
group (Mm, ∆m). This is the “double crossed product” (in the sense of Baaj
and Vaes [3]) of (M 1 , ∆ 1 ) and (M 2 , ∆ 2 ), and is to be called in Definition 3.
below as the dual of the quantum double.
Definition 3.1. Let (N, ∆N ) be a locally compact quantum group, with
WN (“left regular representation”) and VN (“right regular representation”)
being the associated multiplicative unitary operators. In addition, denote
by J N
J
N
, S
N
, ϕ N
, ... the relevant structure maps.
Let (M 1
1
) = (N, ∆
cop
N
), with the multiplicative unitary W 1
= ΣV
∗
N
We have: J 1 = JN and
J 1 =
JN. Also V 1 = (
J 1 ⊗
J 1 )ΣW
∗
1
J 1 ⊗
J 1 ).
Since J
2
1
J
2
1
= I
H
, it becomes: V 1
= ΣW
∗
N
Σ. Meanwhile, let (M 2
2
10 BYUNG-JAY KAHNG
Proposition 3.2. As a von Neumann algebra, we have:
N
D
= N ⊗
N ,
while the comultiplication
D
N
D
N
D
N
D
is characterized as follows:
D
= (id ⊗σ ◦ m ⊗ id)(∆
cop
1
2
) = (id ⊗σ ◦ m ⊗ id)(∆ ⊗
Here σ : N ⊗
N →
N ⊗ N is the flip map, and m : N ⊗
N → N ⊗
N is the
twisting map defined by m(z) = ZzZ
∗ .
Its C
∗ -algebraic counterpart is rather tricky to describe. In general, unless
W
D
is regular (in the sense of Baaj and Skandalis [2]), it may be possible
that
A
D
= A ⊗
A. See discussion given in Section 9 of [3]. Meanwhile, the
description of the comultiplication
∆D given above enables us to prove the
following Lemma, which will be useful later:
Lemma 3.3. Let W = WN ,
W = ΣW
∗ Σ, Z be the operators defined earlier.
Then we have:
Z 34 Z
∗
12
W 24 Z 12
W 13 =
W 13 Z
∗
12
W 24 Z 12 Z 34.
Proof. Since W m
∈ N
D
N
D
is the multiplicative unitary operator giving
rise to the comultiplication
D
, we should have (see [2]):
∆D ⊗ id)(Wm) = Wm, 13 Wm, 23. (3.5)
From the definition of Wm given in equation (3.3), the right side becomes:
Wm, 13 Wm, 23 = W 15 Z
∗
56
W 26 Z 56 W 35 Z
∗
56
W 46 Z 56.
Meanwhile, remembering that
∆(b) =
W
∗ (1 ⊗ b)
W (for b ∈
A) and that
∆(a) = W
∗ (1 ⊗ a)W (for a ∈ A), we have:
∆ ⊗ id)(W m
∆ ⊗ id)
[
W
13
Z
∗
34
W
24
Z
34
]
[
W
∗
12
W
25
W
12
]
Z
∗
56
[
W
∗
34
W
46
W
34
]
Z
56
= W
15
W
25
Z
∗
56
W
36
W
46
Z
56
= W
15
W
25
[Z
∗
56
W
36
Z
56
][Z
∗
56
W
46
Z
56
].
In the third equality, we used the pentagon relations for W and for
W (being
multiplicative unitaries). So we have:
D
⊗ id)(W m
(id ⊗σ ◦ m ⊗ id)(∆ ⊗
(W
m
= Z
32
W
15
W
35
[Z
∗
56
W
26
Z
56
][Z
∗
56
W
46
Z
56
]Z
∗
32
= W
15
Z
32
W
35
[Z
∗
56
W
26
Z
56
]Z
∗
32
[Z
∗
56
W
46
Z
56
].
Therefore, the equation (3.5) now becomes (after obvious cancellations and
then multiplying Z
∗
32
to both sides):
W
35
[Z
∗
56
W
26
Z
56
]Z
∗
32
= Z
∗
32
Z
∗
56
W
26
Z
56
W
35
Re-numbering the legs (legs 2,3,5,6 to become 4,3,1,2), we have:
W
31
Z
∗
12
W
42
Z
12
Z
∗
34
= Z
∗
34
Z
∗
12
W
42
Z
12
W
31
QUANTUM DOUBLE AND THE WEYL ALGEBRA 11
Now taking the adjoints from both sides, it becomes:
Z 34 Z
∗
12
W
∗
42
Z 12 W
∗
31
= W
∗
31
Z
∗
12
W
∗
42
Z 12 Z 34.
Since
W = ΣW
∗ Σ, the result of Lemma follows immediately.
Let us now turn our attention to (N D
D
). We will give a more concrete
realization of N D
(in Proposition 3.4), as well as its coalgebra structure (in
Proposition 3.5). See also Theorem 5.3 of [3].
Proposition 3.4. Define π : N → B(H ⊗ H) and π
′ :
N → B(H ⊗ H) by
π(f ) := Z
∗
(1 ⊗ f )Z and π
′
(k) := k ⊗ 1.
Then N D
is the von Neumann algebra generated by the operators π(f )π
′ (k),
for f ∈ N , k ∈
N. The maps π and π
′ are in fact W
∗ -algebra homomor-
phisms. Namely,
π : N → N D
and π
′
:
N → N
D
Proof. Recall from equation (3.4) that W D
= Z
∗
12
W
24
Z
12
W
13
. So for ω, ω
′ ∈
B(H)
∗
, we have:
(id ⊗ id ⊗ω ⊗ ω
′ )(W D
) = (id ⊗ id ⊗ω ⊗ ω
′ )(Z
∗
12
W
24
Z
12
W
13
= Z
∗
[
1 ⊗ (id ⊗ω
′ )(W )
]
Z
[
(id ⊗ω)(
W ) ⊗ 1
]
= π(f )π
′ (k),
where f = (id ⊗ω
′ )(W ) and k = (id ⊗ω)(
W ). This makes sense, because
W ∈ N ⊗
N and
W ∈
N ⊗ N. Recall the discussion in Section 2 above or
Proposition 2.15 of [11]. In fact, the operators (id ⊗ω
′ )(W ), ω
′ ∈ B(H)∗,
generate the von Neumann algebra N ; while the operators (id ⊗ω)(
W ), ω ∈
B(H)
∗
, generate
N.
Since the operators (id ⊗ id ⊗ω ⊗ ω
′ )(W D
) generate N D
by Definition 3.1,
the claim of the proposition is proved. The second part of the proposition
is obvious from the definitions.
Remark. For future computation purposes, we will from now on regard N D
to be the von Neumann algebra generated by the operators π
′ (k)π(f ), (for
f ∈ M , k ∈
M ). This is of course true, given the results of the previous
proposition. To be more specific, write:
Π(k ⊗ f ) := π
′ (k)π(f ), f ∈ N, k ∈
N. (3.6)
Then we have: ND =
Π(k ⊗ f ) : f ∈ N, k ∈
N
}w
. Its C
∗ -algebraic coun-
terpart is: A D
Π(k ⊗ f ) : f ∈ A, k ∈
A
Proposition 3.5. For f ∈ N and k ∈
N , we have:
∆D
Π(k ⊗ f )
= ∆D
π
′
(k)π(f )
[
(π
′
⊗ π
′
)
∆k
)][
(π ⊗ π)(∆f )
]
k (1)
⊗ f (1)
⊗ k (2)
⊗ f (2)
QUANTUM DOUBLE AND THE WEYL ALGEBRA 13
Consider now ϕ D
given in (1). To verify the left invariance, recall Propo-
sition 3.5 and compute:
(Ω ⊗ ϕ D
D
(Π(k ⊗ f ))
(Ω ⊗ ϕ D
(Π ⊗ Π)(k (1)
⊗ f (1)
⊗ k (2)
⊗ f (2)
∑[
π
′
(k (1)
)π(f (1)
ϕ D
π
′
(k (2)
)π(a (2)
)]
∑[
(k (1)
⊗ 1)Z
∗
(1 ⊗ f (1)
)Z
ϕˆ(k (2)
)ϕ(a (2)
]
Remembering the left invariance property of ϕ, which says: ϕ
(ω⊗id)(∆f )
∑[
ω(f (1)
)ϕ(f (2)
]
= ω(1)ϕ(f ), and similarly for ˆϕ, we thus have:
(Ω ⊗ ϕD)
∆D(Π(k ⊗ f ))
= Ω(1 ⊗ 1) ˆϕ(k)ϕ(f ) = Ω(1 ⊗ 1)ϕD
Π(k ⊗ f )
which is none other than the left invariance property for ϕ D
. Though our
proof is done only at the dense subalgebra level consisting of the Π(k ⊗ f ), it
is sufficient, since we already know the existence of the unique Haar weight
from the general theory. By uniqueness, ϕ D
described here must be the dual
Haar weight on (N D
D
) corresponding to ϕ̂ D
Since we are not going to be prominently using them in this paper, we will
skip the discussions on the right Haar weights and the antipode maps. But
let us just remind the reader that the antipode map SD can be obtained
using the characterization given in equation (2.1), and similarly for
SD,
working now with the operator WD instead.
- The twisting of the quantum double
As is the case in the purely algebraic setting of QUE algebras [6], [4],
the quantum double (A D
D
) or (N D
D
) is equipped with a “quantum
universal R-matrix” type operator R. Our plan is to use this operator to
“twist (deform)” the comultiplication ∆D.
Let us begin by giving the definition and the construction of R, in the op-
erator algebra setting. The approach is more or less the same as in Section 6
of [7], which was in turn adopted from Section 8 of [2]. On the other hand,
some modifications were necessary, because the current situation is more
general than those in [2] and in [7], where the discussions were restricted to
the case of so-called “Kac systems”. At present, the proof here seems to be
the one that is being formulated in the most general setting.
Lemma 4.1. Let W ,
W , Z be the operators in B(H ⊗ H) defined earlier.
Then we have:
(1) Z
∗
12
W 45 W 25 Z 12
W 14 =
W 14 Z
∗
12
W 25 W 45 Z 12
W
35
W
15
Z
∗
34
W
14
Z
34
= Z
∗
34
W
14
Z
34
W
15
W
35
Proof. Recall from Lemma 3.3 that Z 34 Z
∗
12
W 24 Z 12
W 13 =
W 13 Z
∗
12
W 24 Z 12 Z 34
or Z
∗
12
Z
34
W
24
Z
12
W
13
W
13
Z
∗
12
W
24
Z
34
Z
12
. Recall now the definition of
the operator Z given in equation (3.2), and write: Z = W T , where T =
14 BYUNG-JAY KAHNG
JJ ⊗
JJ)W
∗ (
JJ ⊗ J
J). Then our equation above becomes (by writing
Z 34 = W 34 T 34 ):
Z
∗
12
W 34 T 34 W 24 Z 12
W 13 =
W 13 Z
∗
12
W 24 W 34 T 34 Z 12.
We can cancel out T 34 from both sides, because T 34 actually commutes with
all the operators in the equation. To see this, note that
JJ = kJ
J, for a
constant k (actually k = ν
1 / 4 , where ν > 0 is the “scaling constant” [10],
[11]). So we can write:
T = k(J ⊗
J)(
J ⊗ J)W
∗
(
J ⊗ J)(J ⊗
J).
Since W
∗ ∈ M ⊗
M , we can see that T ∈ M
′ ⊗
M
′ .
So far we have: Z
∗
12
W
34
W
24
Z
12
W
13
W
13
Z
∗
12
W
24
W
34
Z
12
. By re-numbering
the legs (letting 3,4 to become 4,5), we obtain (1):
Z
∗
12
W
45
W
25
Z
12
W
14
W
14
Z
∗
12
W
25
W
45
Z
12
Next, we re-write (1), using
W = ΣW
∗ Σ. Then we have:
Z
∗
12
W
∗
54
W
∗
52
Z 12
W 14 =
W 14 Z
∗
12
W
∗
52
W
∗
54
Z 12.
Apply Z
∗
12
W
52
W
54
Z
12
[ · · · ]Z
∗
12
W
54
W
52
Z
12
to both sides. Then:
W
14
Z
∗
12
W
54
W
52
Z
12
= Z
∗
12
W
52
W
54
Z
12
W
14
which is same as:
W 14
W 54 Z
∗
12
W 52 Z 12 = Z
∗
12
W 52 Z 12
W 54
W 14. Here, we re-
number the legs (letting 1,2,4,5 to become 3,4,5,1), and obtain (2):
W 35
W 15 Z
∗
34
W 14 Z 34 = Z
∗
34
W 14 Z 34
W 15
W 35.
Lemma 4.1 above will be helpful in our proof of the next proposition,
which gives the description of our “quantum R-matrix” type operator R.
Proposition 4.2. Let R ∈ B
(H ⊗ H) ⊗ (H ⊗ H)
be the operator defined
by R = Z
∗
34
W
14
Z
34
. The following properties hold:
(1) R ∈ M (A
D
⊗ A
D
) ⊆ N
D
⊗ N
D
and R is unitary: R
− 1 = R
∗ .
(2) We have: (∆D ⊗ id)(R) = R 13 R 23 and (id ⊗∆D)(R) = R 13 R 12.
(3) For any x ∈ AD, we have: R
∆D(x)
R
∗ = ∆
cop
D
(x).
(4) The operator R satisfies the “quantum Yang-Baxter equation (QYBE)”:
Namely, R 12 R 13 R 23 = R 23 R 13 R 12.
Proof. Here M (B) denotes the multiplier algebra of a C
∗ -algebra B.
(1) Recall that
W ∈
N ⊗ N. Therefore, by naturally extending the
∗
homomorphisms π and π
′ defined in Proposition 3.4, we can see that R =
(π
′ ⊗ π)(
W ) ∈ N
D
⊗ N
D
. Actually, noting that
W ∈ M (
A ⊗ A), we also see
that R ∈ M (A D
⊗ A
D
). Meanwhile, from the definitions of the operators
involved, it is clear that R is unitary.
16 BYUNG-JAY KAHNG
using Lemma 4.1 (1). In the seventh and eighth equalities, note
W = ΣW
∗ Σ
and also note ∆
cop (a) = ΣW
∗ (1 ⊗ a)W Σ =
W (a ⊗ 1)
W
∗ .
Since it has been observed that AD is generated by the operators π
′ (b)π(a),
we conclude from the two results above (as well as the unitarity of R) that:
R
[
∆D(x)
]
R
∗ = ∆
cop
D
(x), for any x ∈ AD.
(4) The QYBE follows right away from (3) and (4). In fact,
R 12 R 13 R 23 = R 12
[
(∆D ⊗ id)(R)
]
[
cop
D
⊗ id)(R)
]
R 12 = R 23 R 13 R 12.
The first equality follows from (2); the second equality is from (3); and the
third equality is from (2), with the legs 1 and 2 interchanged.
As a quick consequence of Proposition 4.2, we point out that R determines
a certain “left 2-cocycle” (dual to the notion of same name in the Hopf
algebra setting, introduced in Section 3 of [12]). While we do not need to
give the definition of a 2-cocycle here, this means that we can deform (or
twist) the comultiplication ∆ D
by multiplying R from the left, and obtain
a new map satisfying the coassociativity. The result is given below:
Proposition 4.3. Let R
∆ : A
D
→ M (A
D
⊗ A
D
) be defined by
R
∆(x) := R∆ D
(x), for x ∈ A D
Then R
∆ satisfies the coassociativity: ( R
∆ ⊗ id) R
∆ = (id ⊗ R
R
Proof. The definition for R
∆ makes sense, since R ∈ M (A D
⊗ A
D
). Now
for any x ∈ A D
, we have:
R
∆ ⊗ id) R
∆(x) = R 12
D
⊗ id)
R∆
D
(x)
= R 12
[
D
⊗ id)(R)
][
D
⊗ id)
D
(x)
)]
= R 12
[
R 13 R 23
][
(∆D ⊗ id)
∆D(x)
)]
= R 23
[
R 13 R 12
][
(id ⊗∆D)
∆D(x)
)]
= R 23
[
(id ⊗∆D)(R)
][
(id ⊗∆D)
∆D(x)
)]
= R
23
(id ⊗∆ D
R∆
D
(x)
= (id ⊗ R
R
∆(x).
In the second and sixth equalities, we used the fact that ∆ D
is a C
∗
homomorphism. The third and fifth equalities used Proposition 4.2 (2). In
the fourth equality, we used the QYBE and the coassociativity of ∆ D
The coassociative map (^) R∆ above is certainly a “deformed ∆D”. However,
it should be noted that (AD, (^) R∆) is not going to give us any valid quantum
group. For instance, it is impossible to define a suitable Haar weight. And,
R
∆ is not even a
∗ -homomorphism. On the other hand, considering that
D
is “dual” to the algebra structure on
A
D
(via W D
and Proposition 2.2),
and since R
∆ still carries a sort of a “non-degeneracy” (since ∆ D
is a non-
degenerate C
∗ -morphism and R is a unitary map), we may try to deform
the algebra structure on
A
D
by dualizing R
∆. Formally, we wish to define
QUANTUM DOUBLE AND THE WEYL ALGEBRA 17
on the vector space
A
D
a new product × R
, given by
〈f × R
g | x〉 =
f ⊗ g | R
∆(x)
where f, g ∈
AD and x ∈ AD.
The obvious trouble with this program is that (AD, (^) R∆) is no longer a
quantum group, which means that we do not have any multiplicative unitary
operator that was essential in formulating the dual pairing in the case of
locally compact quantum groups. In the next two sections, we will try to
make sense of the formal equation (4.1), and use it to construct a C
∗ -algebra
(though not a quantum group) that can be considered as a “deformed
A
D
Let us begin with the case of A = C
∗
red
(G).
- The case of an ordinary group. The Weyl algebra.
For this section, let G be an ordinary locally compact group, with a fixed
left Haar measure dx. Let ∇(x) denote the modular function. Using the
Haar measure, we can form the Hilbert space H = L
2 (G). We then construct
two natural subalgebras, N and
N of B(H), as follows.
First consider the von Neumann algebra N = L(G), given by the left
regular representation. That is, for a ∈ C c
(G), let L a
∈ B(H) be such that
L
a
ξ(t) =
a(z)ξ(z
− 1 t) dz. We take L(G) to be the W
∗ -closure of L
C
c
(G)
Next consider
N = L
∞ (G), where b ∈ L
∞ (G) is viewed as the multiplication
operator μ b
on H = L
2 (G), by μ b
ξ(t) = b(t)ξ(t). These are well-known von
Neumann algebras, and it is also known that we can give (mutually dual)
quantum group structures on them. We briefly review the results below.
Let W ∈ B(H ⊗ H) = B
L
2 (G × G)
be defined by W ξ(s, t) = ξ(ts, t). It
is actually the dual (that is, W = ΣW
∗
G
Σ) of the well-known multiplicative
unitary operator WG, defined by WGξ(s, t) = ξ(s, s
− 1 t), and is therefore
multiplicative [2]. We can show without difficulty that
N = L(G) =
(id ⊗ω)(W ) : ω ∈ B(H)∗
}w
and the comultiplication on N is given by ∆(x) = W
∗ (1 ⊗ x)W , for x ∈ N.
For a ∈ Cc(G), this reads: (L ⊗ L)∆aξ(s, t) =
a(z)ξ(z
− 1 s, z
− 1 t) dz. The
antipode map S : a → S(a) is such that
S(a)
(t) = ∇(t
− 1 )a(t
− 1 ), where
∇ is the modular function. The left Haar weight is given by ϕ(a) = a(1),
where 1 = 1 G
is the group identity element. In this way, we obtain a von
Neumann algebraic quantum group (N, ∆), which is co-commutative.
Meanwhile, we can also show that:
N = L
∞
(G) =
(ω ⊗ id)(W ) : ω ∈ B(H)∗
}w
and the comultiplication on
N is given by
∆(y) = ΣW (y⊗1)W
∗ Σ, for y ∈
N.
In effect, this will give us
∆b(s, t) = b(st), for b ∈ L
∞ (G). The antipode
map
S : b →
S(b) is such that
S(b)
(t) = b(t
− 1 ), while the left Haar weight
is just ˆϕ(b) =
b(t) dt. In this way, (
N ,
∆) becomes a commutative von
Neumann algebraic quantum group.
QUANTUM DOUBLE AND THE WEYL ALGEBRA 19
Proof. Recall equation (3.6) for the definition of Π, given in terms of the
∗ -homomorphisms π
′ and π from Proposition 3.4. For (1), note that:
Π(μk ⊗ Lf )ξ(s, t) = π
′
(μk)π(Lf )ξ(s, t) = (μk ⊗ 1)Z
∗
(1 ⊗ Lf )Zξ(s, t)
∇(z)k(s)f (z)ξ(z
− 1 sz, z
− 1 t) dz. (5.2)
If we write α z
ξ(s) = ξ(z
− 1 sz), z ∈ G, as the conjugation action, we can
see without much difficulty from above that the C
∗ -algebra D(G), which is
generated by the operators Π(μ k
⊗ L
f
), is isomorphic to the crossed product
algebra C 0
(G) o α
G. [See any standard textbook on C
∗ -algebras, which
contains discussion on crossed products.] By Proposition 3.5, we also know
that the comultiplication on D(G) is given as in (2).
In our case, being “regular”, we do have:
D(G) = A ⊗
A. At the level of
the functions in Cc(G × G), the multiplication on
D(G) = C
∗
red
(G) ⊗ C 0 (G)
noted in (3) is reflected as follows:
[
(a ⊗ b) × (a
′
⊗ b
′
)
]
(s, t) =
a(z)b(t)a
′
(z
− 1
s)b
′
(t) dsdt. (5.3)
The description given in (4) of the comultiplcation
D
follows from Propo-
sition 3.2.
The next proposition describes the dual pairing map. We may use equa-
tion (2.3), but we instead give our proof using Theorem 2.3.
Proposition 5.2. The dual pairing map is defined between the (dense) sub-
algebras (L ⊗ μ)
C
c
(G × G)
D(G) and Π
(μ ⊗ L)
C
c
(G × G)
⊆ D(G).
Applying Theorem 2.3, we have:
L
a
⊗ μ b
| Π(μ k
⊗ L
f
= (ϕ D
⊗ ϕ̂ D
[
(Π(μ k
⊗ L
f
) ⊗ 1 ⊗ 1)W
∗
D
(1 ⊗ 1 ⊗ L
a
⊗ μ b
]
∇(t)a(t
− 1
st)b(t
− 1
)k(s)f (t) dsdt,
where L a
, L
f
∈ L
C
c
(G)
⊆ A and μ b
, μ k
∈ μ
C
c
(G)
A.
Proof. Recall from Proposition 3.6 that the Haar weights ϕ D
and ϕ̂ D
are
given by
ϕD
Π(μk ⊗ Lf )
= ˆϕ(μk)ϕ(Lf ) =
k(s)f (1) ds,
ϕ̂ D
(L
a
⊗ μ b
) = ϕ(L a
ψ(μ b
) = ϕ(L a
) ˆϕ
S(μ b
a(1)b(t
− 1
) dt.
Meanwhile, remembering the definitions of Π and WD, we have:
(Π(μ k
⊗ L
f
) ⊗ 1 ⊗ 1)W
∗
D
(1 ⊗ 1 ⊗ L
a
⊗ μ b
)ξ(s, t, s
′ , t
′ )
∇(z)∇(t
′
)k(s)f (z)a(z
′
)b(t
′
)ξ(t
′− 1
z
− 1
szt
′
, t
′− 1
z
− 1
t, z
′− 1
z
− 1
szs
′
, t
′
) dzdz
′
.
20 BYUNG-JAY KAHNG
By change of variables (first z
′ 7 → z
− 1 szz
′ , and then z 7 → zt
′− 1 ), it becomes:
∇(zt
′− 1
)k(s)f (zt
′− 1
)a(t
′
z
− 1
szt
′− 1
z
′
)b(t
′
)ξ(z
− 1
sz, z
− 1
t, z
′− 1
s
′
, t
′
) dzdz
′
∇(z)F (s, z, s
′ , t
′ )ξ(z
− 1 sz, z
− 1 t, z
′− 1 s
′ , t
′ ) dzdz
′
([
Π ⊗ (L ⊗ μ)
]
(F )
ξ(s, t, s
′ , t
′ ),
where F (s, z; z
′ , t
′ ) = ∇(t
′− 1 )k(s)f (zt
′− 1 )a(t
′ z
− 1 szt
′− 1 z
′ )b(t
′ ) ∈ Cc(G × G ×
G × G). Recall equation (5.2). Therefore,
L
a
⊗ μ b
| Π(μ k
⊗ L
f
= (ϕ D
⊗ ϕ̂ D
([
Π ⊗ (L ⊗ μ)
]
(F )
F (s, 1 , 1 , t
− 1 ) dsdt =
∇(t)k(s)f (t)a(t
− 1 st)b(t
− 1 ) dsdt.
By Theorem 2.3, we know that this is a valid dual pairing map (at the
level of dense subalgebras) between
D(G) and D(G), satisfying (1),(2),(3)
of Proposition 2.2. In particular, the property (1) implies that:
(La⊗μ b
)(L
a
′ (^) ⊗μ b
′ (^) ) | Π(μ k
⊗L
f
(La⊗μ b
)⊗(L
a
′ (^) ⊗μ b
D(Π(μk ⊗Lf ))
which relates the comultiplication ∆ D
on D(G) with the product on
D(G).
Even though we expressed our dual pairing as between certain subalgebras
of
D(G) and D(G), note that the pairing map is in effect being considered
at the level of functions in C c
(G × G). In that sense, we may write the
pairing map given in Proposition 5.2 as:
〈a ⊗ b | k ⊗ f 〉 =
∇(t)a(t
− 1 st)b(t
− 1 )k(s)f (t) dsdt. (5.4)
Let us now consider the deformed comultiplication R
∆ proposed in the
previous section, and by using the dual pairing, try to “deform” the algebra
C
∗ (G) ⊗ C 0
(G). Since the dual pairing is valid only at the level of functions,
we will first work in the subspace C c
(G × G). Formally, we wish to deform
its product given in equation (5.3) to a new one, so that the new product is
“dual” to R
∆, as suggested by equation (4.1). In our case, we look for the
“deformed product” × R
, satisfying (formally) the following:
[(a ⊗ b) × R
(a
′
⊗ b
′
)] | k ⊗ f
(a ⊗ b) ⊗ (a
′
⊗ b
′
) | R
∆(k ⊗ f )
To make some sense of this, we first need to regard R
∆(k⊗f ) as a (general-
ized) function on G×G. So consider k, f ∈ C c
(G), and consider Π(μ k
⊗L
f
D(G). By definition, and by remembering that R = Z
∗
34
W 14 Z 34 , we have:
R
Π(μ k
⊗ L
f
ξ(s, t, s
′
, t
′
) = R∆ D
Π(μ k
⊗ L
f
ξ(s, t, s
′
, t
′
)
= ∇(s)W
∗
D
1 ⊗ 1 ⊗ Π(μ k
⊗ L
f
W
D
ξ(s, t, s
− 1 s
′ s, s
− 1 t
′ )
∇(s)∇(z)∇(z)k(s
′
s)f (z)ξ(z
− 1
sz, z
− 1
t, z
− 1
s
− 1
s
′
sz, z
− 1
s
− 1
t
′
) dz.
