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 :
The comultiplication is not necessarily unit–preserving.
The counit is not necessarily a homomorphism of algebras.
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
along with the counit axiom:
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
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
where, for , the twisted tensor product is specified by
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
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 :
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 .
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
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).
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.
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 denote a complex (separable) Hilbert space. A von Neumann algebra acting on is a subset of the algebra of all bounded operators such that:
is closed under the adjoint operation (with the adjoint of an element denoted by ).
equals its bicommutant, namely:
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).
1.2 Definition of a Hopf algebra
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 :
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
There is also a counterpart to , the counity map satisfying
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 :
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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|Title||weak Hopf algebra|
|Date of creation||2013-03-22 18:12:43|
|Last modified on||2013-03-22 18:12:43|
|Last modified by||bci1 (20947)|
|Synonym||quantum groupoids v.1|
|Defines||commutant of a set|
|Defines||twisted tensor product|