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
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:
![$\displaystyle \xymatrix@C=3pc@R=3pc{ A \otimes A \ar[d]_{\varepsilon \otimes 1}...
...{\rm id}_A} \ar[d]^{\Delta} \\ A & A \otimes A \ar[l]^{1 \otimes \varepsilon }}$](img6.png) |
(0.3) |
Some authors substitute the term quantum `groupoid' for a weak Hopf algebra.
- (1)
- We refer here to Bais et al. (2002). Let
be a non-Abelian
group
and
a discrete subgroup. Let
denote the space
of functions
on
and
the group algebra (which consists
of the linear span of group elements with the group structure).
The quantum double
(Drinfeld, 1987) is defined by
 |
(0.4) |
where, for
, the `twisted tensor
product' is specified by
 |
(0.5) |
The physical interpretation is often to take
as the `electric gauge group' and
as the `magnetic symmetry' generated by
. In terms of the counit
, the double
has a trivial representation
given by
. We next look at certain features of this construction.
For the purpose of braiding relations
there is an
matrix,
, leading to the operator
 |
(0.6) |
in terms of the Clebsch-Gordan series
, and
where
denotes a flip operator. The operator
is sometimes called the monodromy or
Aharanov-Bohm phase factor. In the case of a condensate in
a state
in the carrier space of some
representation
. One considers the maximal Hopf
subalgebra
of a Hopf algebra
for which
is
-invariant; specifically :
 |
(0.7) |
- (2)
- For the second example, consider
. The algebra of
functions on
can be broken to the algebra of functions on
, that is, to
, where
is normal in
, that is,
. Next, consider
. On breaking a purely
electric condensate
, the magnetic symmetry
remains unbroken, but the electric symmetry
is broken to
, with
, the stabilizer of
. From this we obtain
.
- (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
 |
(0.8) |
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 2 in the
Jones extension. Related is how a von Neumann algebra
, such as
of finite index depth 2, sits inside a weak Hopf algebra formed as
the crossed product
(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
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 us recall two basic concepts
of quantum operator algebra
that are essential to algebraic
quantum 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:
 |
(1.1) |
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 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).
Firstly, a unital associative algebra consists of a linear space
together with two linear maps
satisfying the conditions
This first condition can be seen in terms of a commuting diagram :
 |
(1.4) |
Next suppose we consider `reversing the arrows', and take an
algebra
equipped with a linear homorphisms
, satisfying, for
:
We call
a comultiplication, which is said to be
coasociative in so far that the following diagram commutes
 |
(1.6) |
There is also a counterpart to
, the counity map
satisfying
 |
(1.7) |
A bialgebra
is a linear space
with maps
satisfying the above properties.
Now to recover anything resembling a group structure, we must
append such a bialgebra with an antihomomorphism
,
satisfying
, for
. This map is
defined implicitly via the property :
 |
(1.8) |
We call
the antipode map. A Hopf algebra is then
a bialgebra
equipped with an antipode
map
.
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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