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Study Notes on Quantum Double and its Representation, Papers of Introduction to Sociology

These study notes explore the concept of quantum double, focusing on its representation in the context of hopf algebras. Topics such as the pentagon equation, the co-opposite comultiplication, the weight and its right invariance, and the construction of the multiplicative unitary operator wm. It also discusses the von neumann algebra generated by certain operators and its comultiplication.

Typology: Papers

Pre 2010

Uploaded on 08/18/2009

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bg1
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 Rdetermines 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 C0(G)×τG, which is in turn isomorphic to K(L2(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.
1. 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- or von Neumann al-
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 Ais associated with a certain dual object ˆ
A, which
is also a quantum group, and the dual object, ˆ
ˆ
A, of the dual quantum group
is actually isomorphic to A. This result, ˆ
ˆ
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 H0, with its Hopf algebra structure obtained naturally
1
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d

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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.

  1. 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

  • or von Neumann al-

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.

  1. 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.

  1. 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

)

∆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.

  1. 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)(

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).

  1. 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.