weak Hopf C*-algebra
Definition 0.1
A
weak Hopf
-algebra is defined as a weak Hopf algebra 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. 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”

.
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
is not necessarily a homomorphism
of algebras.
- (3)
- The axioms for the antipode map
with respect to the
counit are as follows. For all
,
These axioms may be appended by the following commutative diagrams
 |
(0.2) |
along with the counit axiom:
![$\displaystyle \xymatrix@C=3pc@R=3pc{ A \otimes A \ar[d]_{\varepsilon \otimes 1}...
...{\rm id}_A} \ar[d]^{\Delta} \\ A & A \otimes A \ar[l]^{1 \otimes \varepsilon }}$](img9.png) |
(0.3) |
Some authors substitute the term quantum `groupoid' for a weak Hopf 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
 |
(0.4) |
is the Jones extension induced by a finite index depth
inclusion
of
factors, then
admits a quantum groupoid structure and acts on
, so that
and
. 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
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
in the
Jones extension. Related is how a von Neumann algebra
, such as
of finite index depth
, sits inside a weak Hopf algebra formed as
the crossed product
(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
with
. If
, then it is shown that a QTQHA is
canonically associated with
. Such QTQHAs are
claimed as the true symmetries of minimal conformal field
theories.
Let
denote a complex (separable) Hilbert space. A von
Neumann algebra
acting on
is a subset of the
-algebra of
all bounded operators
such that:
- (1)
-
is closed under the adjoint operation
(with the
adjoint of an element
denoted by
).
- (2)
-
equals its bicommutant, namely:
 |
(0.5) |
If one calls a commutant of a set
the special set of
bounded operators on
which commute
with all elements in
, then this second condition implies that the commutant of the
commutant of
is again the set
.
On the other hand, a von Neumann algebra
inherits a
unital subalgebra from
, and according to the
first condition in its definition
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
is a von
Neumann algebra if and only if
is a *-subalgebra of
, closed for the smallest topology defined by continuous
maps
for all
where
denotes the inner product
defined on
. 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.
-
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