weak Hopf C*-algebra

weak Hopf C*-algebra

Definition 0.1

.

A

weak Hopf

C

*

-

algebra

is defined as a

weak Hopf algebra

(

http://planetmath.org/WeakHopfCAlgebra

) which admits a

faithful

*

–representation on a

Hilbert space

. The weak C*–

Hopf algebra

is therefore much more likely to be closely related to a

quantum groupoid

than the weak Hopf algebra. However, one can argue that

locally compact groupoids

equipped with a Haar measure are even closer to defining quantum groupoids (

http://planetmath.org/QuantumGroupoids2

).

There are already several, significant examples that motivate the consideration of weak C*-Hopf algebras which also deserve mentioning in the

context

of standard

quantum theories

. Furthermore, notions such as (proper)

weak C*-algebroids

can provide the main framework for symmetry breaking and

quantum gravity

that we are considering here. Thus, one may consider the quasi-group

symmetries

constructed by means of special

transformations

of the coordinate space

M

.

Remark

:
Recall that the weak Hopf algebra is defined as the

extension

of a Hopf algebra by weakening
the definining 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.

0.1

Examples of weak Hopf C*-algebra.

(1)

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

noncommutative

symmetries of depth 2 von Neumann subfactors. If

A

⊂

B

⊂

B

1

⊂

B

2

⊂

…

(0.4)

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).

(2)

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.

0.2

Von Neumann Algebras (or

W

*

-algebras).

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

)

}

.

(0.5)

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 the 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).

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

1

E. M. Alfsen and F. W. Schultz:

Geometry of

State Spaces

of Operator Algebras

, Birkhäuser, Boston–Basel–Berlin (2003).

2

I. Baianu :

Categories

,

Functors

and Automata Theory: A Novel Approach to

Quantum Automata

through

Algebraic

–Topological Quantum Computations.,

Proceed. 4th Intl. Congress LMPS

, (August-Sept. 1971).

3

I. C. Baianu, J. F. Glazebrook and R. Brown.: A Non–

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

Algebraic Topology

.

in preparation

, (2008).

5

F.A. Bais, B. J. Schroers and J. K. Slingerland: Broken quantum symmetry and confinement phases in planar physics,

Phys. Rev. Lett.

89

No. 18 (1–4): 181–201 (2002).

6

M. R. Buneci.:

Groupoid Representations

, Ed. Mirton: Timishoara (2003).

7

M. Chaician and A. Demichev:

Introduction

to Quantum Groups

, World Scientific (1996).

8

L. Crane and I.B. Frenkel. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. Topology and physics.

J. Math. Phys

.

35

(no. 10): 5136–5154 (1994).

9

V. G. Drinfel’d: Quantum groups, In

Proc. Intl. Congress of
Mathematicians, Berkeley 1986

, (ed. A. Gleason), Berkeley, 798-820 (1987).

10

G. J. Ellis: Higher dimensional crossed modules of algebras,

J. of Pure Appl. Algebra

52

(1988), 277-282.

11

P.. I. Etingof and A. N. Varchenko,

Solutions

of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups,

Comm.Math.Phys.

,

196

: 591-640 (1998).

12

P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups,

Commun. Math. Phys.

205

(1): 19-52 (1999)

13

P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang–Baxter equations, in

Quantum Groups and Lie Theory (Durham, 1999)

, pp. 89-129, Cambridge University Press, Cambridge, 2001.

14

B. Fauser:

A treatise on quantum Clifford Algebras

. Konstanz, Habilitationsschrift.

arXiv.math.QA/0202059 (2002).

15

B. Fauser: Grade

Free product

Formulae from Grassman–Hopf Gebras. Ch. 18 in R. Ablamowicz, Ed.,

Clifford Algebras: Applications to Mathematics, Physics and Engineering

, Birkhäuser: Boston, Basel and Berlin, (2004).

16

J. M. G. Fell.: The

Dual Spaces

of C*–Algebras.,

Transactions of the American
Mathematical Society

,

94

: 365–403 (1960).

17

F.M. Fernandez and E. A. Castro.:

(Lie) Algebraic Methods in Quantum Chemistry and Physics.

, Boca Raton: CRC Press, Inc (1996).

18

R. P. Feynman: Space–Time Approach to Non–Relativistic Quantum Mechanics,

Reviews
of Modern Physics

, 20: 367–387 (1948). [It is also reprinted in (Schwinger 1958).]

19

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Homological Algebra

. I.

Derived functors

and satellites.,

Proc. London Math. Soc.

,

11

(3): 239–252 (1961).

20

R. Gilmore:

Lie Groups

,

Lie Algebras

and Some of Their Applications.

,
Dover Publs., Inc.: Mineola and New York, 2005.

21

P. Hahn: Haar measure for

measure groupoids

.,

Trans. Amer. Math. Soc

.

242

: 1–33(1978).

22

P. Hahn: The

regular representations

of measure groupoids.,

Trans. Amer. Math. Soc

.

242

:34–72(1978).

23

R. Heynman and S. Lifschitz. 1958.

Lie Groups and Lie Algebras

., New York and London: Nelson Press.

24

Leonid Vainerman, Editor. 2003.

https://perswww.kuleuven.be/ u0018768/artikels/strasbourg.pdf

Locally Compact Quantum Groups and Groupoids: Proceedings of the Meeting of Theoretical Physicists and Mathematicians

., Strasbourg, February 21-23, 2002., Walter de Gruyter Gmbh & Co: Berlin.

25

http://planetmath.org/?op=getobj&from=books&id=294

Stefaan Vaes and Leonid Vainerman.2003. On Low-Dimensional Locally Compact
Quantum Groups in

Locally Compact Quantum Groups and Groupoids: Proceedings of the Meeting of Theoretical Physicists and Mathematicians

Title

weak Hopf C*-algebra

Canonical name

WeakHopfCalgebra

Date of creation

2013-03-22 18:12:47

Last modified on

2013-03-22 18:12:47

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

62

Author

bci1 (20947)

Entry type

Topic

Classification

msc 08C99

Classification

msc 16S40

Classification

msc 81R15

Classification

msc 81R50

Classification

msc 16W30

Classification

msc 57T05

Synonym

quantum groupoids

Related topic

WeakHopfAlgebra

Related topic

VonNeumannAlgebra

Related topic

TopologicalAlgebra

Related topic

QuantumGroupoids2

Related topic

LocallyCompactQuantumGroup

Related topic

HopfAlgebra

Related topic

LocallyCompactGroupoids

Related topic

QuantumGroupoids2

Related topic

GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries

Related topic

GrassmanHopfAlgebrasAndTheirDualCoAlge

Defines

weak Hopf algebra

Defines

weak Hopf C*-algebra

Defines

weak bialgebra

Defines

quantum group

Defines

quantum groupoid

Defines

von Neumann algebra

Defines

W

*

–algebra

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