weak Hopf algebra

weak Hopf algebra

Definition 0.1

:
In order to define a

weak Hopf algebra

, one weakens, or relaxes certain axioms of a

Hopf algebra

as follows :

(1)

The comultiplication is not necessarily unit–preserving.

(2)

The counit

ε

is not necessarily a

homomorphism

of

algebras

.

(3)

The axioms for the

antipode map

S

:

A

⟶

A

with respect to the
counit are as follows. For all

h

∈

H

,

m

⁢

(

id

⊗

S

)

⁢

Δ

⁢

(

h

)

=

(

ε

⊗

id

)

⁢

(

Δ

⁢

(

1

)

⁢

(

h

⊗

1

)

)

(0.1)

m

⁢

(

S

⊗

id

)

⁢

Δ

⁢

(

h

)

=

(

id

⊗

ε

)

⁢

(

(

1

⊗

h

)

⁢

Δ

⁢

(

1

)

)

S

⁢

(

h

)

=

S

⁢

(

h

(

1

)

)

⁢

h

(

2

)

⁢

S

⁢

(

h

(

3

)

)

.

These axioms may be appended by the following

commutative diagrams

A

⊗

A

→

S

⊗

id

A

⊗

A

Δ

↑

↓

m

A

⁢

@

>

u

∘

ε

≫

A

A

⊗

A

→

id

⊗

S

A

⊗

A

Δ

↑

↓

m

A

⁢

@

>

u

∘

ε

≫

A

(0.2)

along with the counit axiom:

\xymatrix

⁢

@

⁢

C

=

3

⁢

p

⁢

c

⁢

@

⁢

R

=

3

⁢

p

⁢

c

⁢

A

⊗

A

⁢

\ar

⁢

[

d

]

ε

⊗

1

⁢

&

⁢

A

⁢

\ar

⁢

[

l

]

Δ

⁢

\ar

⁢

[

d

⁢

l

]

id

A

⁢

\ar

⁢

[

d

]

Δ

⁢

A

⁢

&

⁢

A

⊗

A

⁢

\ar

⁢

[

l

]

1

⊗

ε

(0.3)

Some authors substitute the term

quantum groupoid

for a weak Hopf algebra.
Therefore, the weak Hopf algebra is considered by some authors as an important

concept

in

quantum operator algebra

(QOA).

0.1

Examples of weak Hopf algebras

(1)

We refer here to Bais et al. (2002). Let

G

be a

non-Abelian group

and

H

⊂

G

a discrete subgroup. Let

F

⁢

(

H

)

denote the

space
of functions

on

H

and

ℂ

⁢

H

the

group algebra

(which consists
of the

linear span

of group

elements

with the group

structure

).

The

quantum double

D

⁢

(

H

)

(Drinfeld, 1987) is defined by

D

⁢

(

H

)

=

F

⁢

(

H

)

⁢

⊗

~

⁢

ℂ

⁢

H

,

(0.4)

where, for

x

∈

H

, the

twisted tensor product

is specified by

⊗

~

↦

(

f

1

⊗

h

1

)

⁢

(

f

2

⊗

h

2

)

⁢

(

x

)

=

f

1

⁢

(

x

)

⁢

f

2

⁢

(

h

1

⁢

x

⁢

h

1

-

1

)

⊗

h

1

⁢

h

2

.

(0.5)

The physical

interpretation

is often to take

H

as the ‘electric

gauge group

’ and

F

⁢

(

H

)

as the ‘magnetic

symmetry

’ generated by

{

f

⊗

e

}

. In terms of the counit

ε

, the double

D

⁢

(

H

)

has a

trivial representation

given by

ε

⁢

(

f

⊗

h

)

=

f

⁢

(

e

)

. We next look at certain features of this construction.

For the purpose of braiding

relations

there is an

R

matrix,

R

∈

D

⁢

(

H

)

⊗

D

⁢

(

H

)

, leading to the

operator

ℛ

≡

σ

⋅

(

Π

α

A

⊗

Π

β

B

)

⁢

(

R

)

,

(0.6)

in terms of the Clebsch–Gordan series

Π

α

A

⊗

Π

β

B

≅

N

α

⁢

β

⁢

C

A

⁢

B

⁢

γ

⁢

Π

γ

C

, and
where

σ

denotes a flip operator. The operator

ℛ

2

is sometimes called the

monodromy

or

Aharanov–Bohm phase factor

. In the case of a condensate in
a state

|

v

⟩

in the

carrier

space of some

representation

Π

α

A

. One considers the

maximal

Hopf

subalgebra

T

of a Hopf algebra

A

for which

|

v

⟩

is

T

–

invariant

; specifically :

Π

α

A

⁢

(

P

)

⁢

|

v

⟩

=

ε

⁢

(

P

)

⁢

|

v

⟩

,

∀

P

∈

T

.

(0.7)

(2)

For the second example, consider

A

=

F

⁢

(

H

)

. The algebra of
functions on

H

can be broken to the algebra of functions on

H

/

K

, that is, to

F

⁢

(

H

/

K

)

, where

K

is normal in

H

, that is,

H

⁢

K

⁢

H

-

1

=

K

. Next, consider

A

=

D

⁢

(

H

)

. On breaking a purely
electric condensate

|

v

⟩

, the magnetic symmetry
remains unbroken, but the electric symmetry

ℂ

⁢

H

is broken to

ℂ

⁢

N

v

, with

N

v

⊂

H

, the

stabilizer

of

|

v

⟩

. From this we obtain

T

=

F

⁢

(

H

)

⁢

⊗

~

⁢

ℂ

⁢

N

v

.

(3)

In Nikshych and Vainerman (2000) quantum groupoids (as weak
C*–Hopf algebras, see below) were studied in relationship to the

noncommutative

symmetries of depth 2 von Neumann subfactors. If

A

⊂

B

⊂

B

1

⊂

B

2

⊂

…

(0.8)

is the Jones

extension

induced by a finite index depth

2

inclusion

A

⊂

B

of

I

⁢

I

1

factors, then

Q

=

A

′

∩

B

2

admits a quantum groupoid structure and acts on

B

1

, so that

B

=

B

1

Q

and

B

2

=

B

1

⋊

Q

. Similarly, in Rehren (1997)
‘paragroups’ (derived from weak C*–Hopf algebras) comprise
(quantum)

groupoids

of

equivalence classes

such as associated with
6j–

symmetry groups

(relative to a fusion rules algebra). They
correspond to type

I

⁢

I

von Neumann algebras

in quantum mechanics,
and arise as symmetries where the local subfactors (in the sense
of containment of observables within fields) have depth 2 in the
Jones extension. Related is how a von Neumann algebra

N

, such as
of finite index depth 2, sits inside a weak Hopf algebra formed as
the crossed product

N

⋊

A

(Böhm et al. 1999).

(4)

In Mack and Schomerus (1992) using a more general notion of the
Drinfeld construction, develop the notion of a

quasi
triangular quasi–Hopf algebra

(QTQHA) is developed with the aim
of studying a range of essential symmetries with special
properties, such the

quantum group

algebra

U

q

⁢

(

sl

2

)

with

|

q

|

=

1

. If

q

p

=

1

, then it is shown that a QTQHA is
canonically associated with

U

q

⁢

(

sl

2

)

. Such QTQHAs are
claimed as the true symmetries of

minimal

conformal field
theories.

1

Definitions of Related Concepts

Let us recall two basic concepts of quantum operator algebra that are essential to Algebraic

Quantum Theories

.

1.1

Definition of a Von Neumann Algebra.

Let

ℋ

denote a complex (

separable

)

Hilbert space

. A

von
Neumann algebra

𝒜

acting on

ℋ

is a subset of the algebra of
all

bounded operators

ℒ

⁢

(

ℋ

)

such that:

(1)

𝒜

is

closed under

the

adjoint

operation

(with the
adjoint of an element

T

denoted by

T

*

).

(2)

𝒜

equals its bicommutant, namely:

𝒜

=

{

A

∈

ℒ

(

ℋ

)

:

∀

B

∈

ℒ

(

ℋ

)

,

∀

C

∈

𝒜

,

(

B

C

=

C

B

)

⇒

(

A

B

=

B

A

)

}

.

(1.1)

If one calls a

commutant

of a set

𝒜

the special set of
bounded operators on

ℒ

⁢

(

ℋ

)

which commute with all elements in

𝒜

, then this second condition implies that the commutant of the
commutant of

𝒜

is again the set

𝒜

.

On the other hand, a von Neumann algebra

𝒜

inherits a

unital

subalgebra from

ℒ

⁢

(

ℋ

)

, and according to the
first condition in its definition

𝒜

does indeed inherit a

*-subalgebra

structure, as further explained in the next

section

on C*-algebras. Furthermore, we have notable

Bicommutant Theorem

which states that

𝒜

is a von
Neumann algebra if and only if

A

is a *-subalgebra of

L

⁢

(

H

)

, closed for the smallest

topology

defined by

continuous
maps

(

ξ

,

η

)

⟼

(

A

⁢

ξ

,

η

)

for all

<

A

ξ

,

η

)

>

where

<

.

,

.

>

denotes the

inner product

defined on

H

. For
further instruction on this subject, see e.g. Aflsen and Schultz
(2003), Connes (1994).

1.2

Definition of a Hopf algebra

Firstly, a

unital associative algebra

consists of a

linear space

A

together with two

linear maps

m

:

A

⊗

A

⟶

A

,

(

m

⁢

u

⁢

l

⁢

t

⁢

i

⁢

p

⁢

l

⁢

i

⁢

c

⁢

a

⁢

t

⁢

i

⁢

o

⁢

n

)

(1.2)

η

:

ℂ

⟶

A

,

(

u

⁢

n

⁢

i

⁢

t

⁢

y

)

satisfying the conditions

m

⁢

(

m

⊗

𝟏

)

=

m

⁢

(

𝟏

⊗

m

)

(1.3)

m

⁢

(

𝟏

⊗

η

)

=

m

⁢

(

η

⊗

𝟏

)

=

id

.

This first condition can be seen in terms of a commuting diagram :

A

⊗

A

⊗

A

→

m

⊗

id

A

⊗

A

id

⊗

m

↓

↓

m

A

⊗

A

⁢

@

>

m

≫

A

(1.4)

Next suppose we consider ‘reversing the arrows’, and take an
algebra

A

equipped with a linear homorphisms

Δ

:

A

⟶

A

⊗

A

, satisfying, for

a

,

b

∈

A

:

Δ

⁢

(

a

⁢

b

)

=

Δ

⁢

(

a

)

⁢

Δ

⁢

(

b

)

(1.5)

(

Δ

⊗

id

)

⁢

Δ

=

(

id

⊗

Δ

)

⁢

Δ

.

We call

Δ

a

comultiplication

, which is said to be

coasociative

in so far that the following diagram commutes

A

⊗

A

⊗

A

←

Δ

⊗

id

A

⊗

A

id

⊗

Δ

↑

↑

Δ

A

⊗

A

⁢

@

<

Δ

≪

A

(1.6)

There is also a counterpart to

η

, the

counity

map

ε

:

A

⟶

ℂ

satisfying

(

id

⊗

ε

)

∘

Δ

=

(

ε

⊗

id

)

∘

Δ

=

id

.

(1.7)

A

bialgebra

(

A

,

m

,

Δ

,

η

,

ε

)

is a linear space

A

with maps

m

,

Δ

,

η

,

ε

satisfying the above properties.

Now to recover anything resembling a group structure, we must
append such a bialgebra with an

antihomomorphism

S

:

A

⟶

A

,
satisfying

S

⁢

(

a

⁢

b

)

=

S

⁢

(

b

)

⁢

S

⁢

(

a

)

, for

a

,

b

∈

A

. This map is
defined implicitly via the property :

m

⁢

(

S

⊗

id

)

∘

Δ

=

m

⁢

(

id

⊗

S

)

∘

Δ

=

η

∘

ε

.

(1.8)

We call

S

the

antipode map

. A

Hopf algebra

is then
a bialgebra

(

A

,

m

,

η

,

Δ

,

ε

)

equipped with an antipode
map

S

.

Commutative

and noncommutative Hopf algebras form the backbone of
quantum ‘groups’ and are essential to the

generalizations

of
symmetry. Indeed, in most respects a quantum ‘group’ is identifiable
with a Hopf algebra. When such algebras are actually
associated with proper groups of matrices there is
considerable scope for their representations on both finite
and infinite dimensional Hilbert spaces.

References

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Geometry

of

State Spaces

of Operator Algebras

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I. Baianu :

Categories

,

Functors

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Quantum Automata

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3

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Abelian

,

Categorical Ontology

of Spacetimes and

Quantum Gravity

.,

Axiomathes

17

,(3-4): 353-408(2007).

4

I.C.Baianu, R. Brown J.F. Glazebrook, and G. Georgescu, Towards Quantum Non–Abelian

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.

in preparation

, (2008).

5

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89

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6

J.W. Barrett.: Geometrical measurements in three-dimensional quantum gravity.
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A 18

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7

M. R. Buneci.:

Groupoid Representations

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8

M. Chaician and A. Demichev:

Introduction

to Quantum Groups

, World Scientific (1996).

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Phys. Rev. D

21

: 3305 (1980).

10

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.

35

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Hilbert bundle

over spacetime.,

Classical and Quantum Gravity

,

13

:611-632 (1996).
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12

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Proc. Intl. Congress of
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,

196

: 591-640 (1998).

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205

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arXiv:0709.4364v2 [quant–ph]

Title

weak Hopf algebra

Canonical name

WeakHopfAlgebra

Date of creation

2013-03-22 18:12:43

Last modified on

2013-03-22 18:12:43

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

35

Author

bci1 (20947)

Entry type

Definition

Classification

msc 08C99

Classification

msc 81R15

Classification

msc 57T05

Classification

msc 81R50

Classification

msc 16W30

Synonym

quantum groupoids v.1

Related topic

HopfAlgebra

Related topic

WeakHopfCAlgebra2

Related topic

WeakHopfCAlgebra2

Related topic

CommutativeDiagram

Related topic

GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries

Related topic

GrassmanHopfAlgebrasAndTheirDualCoAlgebras

Related topic

WeakHopfCAlgebra2

Defines

weak bialgebra

Defines

commutant of a set

Defines

counit axiom

Defines

antipode map

Defines

counity

Defines

twisted tensor product

Defines

quantum double

Defines

QOA

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