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 $\epsilon $ is not necessarily a homomorphism^{} of algebras^{}.

(3)
The axioms for the antipode map $S:A\u27f6A$ with respect to the counit are as follows. For all $h\in H$,
$m(\mathrm{id}\otimes S)\mathrm{\Delta}(h)$ $=(\epsilon \otimes \mathrm{id})(\mathrm{\Delta}(1)(h\otimes 1))$ (0.1) $m(S\otimes \mathrm{id})\mathrm{\Delta}(h)$ $=(\mathrm{id}\otimes \epsilon )((1\otimes h)\mathrm{\Delta}(1))$ $S(h)$ $=S({h}_{(1)}){h}_{(2)}S({h}_{(3)}).$
These axioms may be appended by the following commutative diagrams^{}
$$\begin{array}{ccc}\hfill A\otimes A\hfill & \hfill \stackrel{S\otimes \mathrm{id}}{\to}\hfill & \hfill A\otimes A\hfill \\ \hfill \mathrm{\Delta}\uparrow \hfill & & \hfill \downarrow m\hfill & & \\ \hfill A\text{@}>u\circ \epsilon \gg A\hfill \end{array}\mathit{\hspace{1em}\hspace{1em}}\begin{array}{ccc}\hfill A\otimes A\hfill & \hfill \stackrel{\mathrm{id}\otimes S}{\to}\hfill & \hfill A\otimes A\hfill \\ \hfill \mathrm{\Delta}\uparrow \hfill & & \hfill \downarrow m\hfill & & \\ \hfill A\text{@}>u\circ \epsilon \gg A\hfill \end{array}$$  (0.2) 
along with the counit axiom:
$$\text{xymatrix}\mathrm{@}C=3pc\mathrm{@}R=3pcA\otimes A\text{ar}{[d]}_{\epsilon \otimes 1}\mathrm{\&}A\text{ar}{[l]}_{\mathrm{\Delta}}\text{ar}{[dl]}_{{\mathrm{id}}_{A}}\text{ar}{[d]}^{\mathrm{\Delta}}A\mathrm{\&}A\otimes A\text{ar}{[l]}^{1\otimes \epsilon}$$  (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 nonAbelian group^{} and $H\subset G$ a discrete subgroup. Let $F(H)$ denote the space of functions on $H$ and $\u2102H$ 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
$$D(H)=F(H)\stackrel{~}{\otimes}\u2102H,$$ (0.4) where, for $x\in H$, the twisted tensor product is specified by
$$\stackrel{~}{\otimes}\mapsto ({f}_{1}\otimes {h}_{1})({f}_{2}\otimes {h}_{2})(x)={f}_{1}(x){f}_{2}({h}_{1}x{h}_{1}^{1})\otimes {h}_{1}{h}_{2}.$$ (0.5) The physical interpretation^{} is often to take $H$ as the ‘electric gauge group’ and $F(H)$ as the ‘magnetic symmetry^{}’ generated by $\{f\otimes e\}$ . In terms of the counit $\epsilon $, the double $D(H)$ has a trivial representation given by $\epsilon (f\otimes h)=f(e)$ . We next look at certain features of this construction.
For the purpose of braiding relations^{} there is an $R$ matrix, $R\in D(H)\otimes D(H)$, leading to the operator
$$\mathcal{R}\equiv \sigma \cdot ({\mathrm{\Pi}}_{\alpha}^{A}\otimes {\mathrm{\Pi}}_{\beta}^{B})(R),$$ (0.6) in terms of the Clebsch–Gordan series ${\mathrm{\Pi}}_{\alpha}^{A}\otimes {\mathrm{\Pi}}_{\beta}^{B}\cong {N}_{\alpha \beta C}^{AB\gamma}{\mathrm{\Pi}}_{\gamma}^{C}$, and where $\sigma $ denotes a flip operator. The operator ${\mathcal{R}}^{2}$ is sometimes called the monodromy^{} or Aharanov–Bohm phase factor. In the case of a condensate in a state $v\u27e9$ in the carrier^{} space of some representation^{} ${\mathrm{\Pi}}_{\alpha}^{A}$ . One considers the maximal Hopf subalgebra^{} $T$ of a Hopf algebra $A$ for which $v\u27e9$ is $T$–invariant^{}; specifically :
$${\mathrm{\Pi}}_{\alpha}^{A}(P)v\u27e9=\epsilon (P)v\u27e9,\forall P\in T.$$ (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, $HK{H}^{1}=K$ . Next, consider $A=D(H)$ . On breaking a purely electric condensate $v\u27e9$, the magnetic symmetry remains unbroken, but the electric symmetry $\u2102H$ is broken to $\u2102{N}_{v}$, with ${N}_{v}\subset H$, the stabilizer^{} of $v\u27e9$ . From this we obtain $T=F(H)\stackrel{~}{\otimes}\u2102{N}_{v}$ .

(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
$$A\subset B\subset {B}_{1}\subset {B}_{2}\subset \mathrm{\dots}$$ (0.8) is the Jones extension^{} induced by a finite index depth $2$ inclusion $A\subset B$ of $I{I}_{1}$ factors, then $Q={A}^{\prime}\cap {B}_{2}$ admits a quantum groupoid structure and acts on ${B}_{1}$, so that $B={B}_{1}^{Q}$ and ${B}_{2}={B}_{1}\u22caQ$ . 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 $II$ 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 $N\u22caA$ (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 ${\mathrm{U}}_{q}({\mathrm{sl}}_{2})$ with $q=1$ . If ${q}^{p}=1$, then it is shown that a QTQHA is canonically associated with ${\mathrm{U}}_{q}({\mathrm{sl}}_{2})$. 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 $\mathscr{H}$ denote a complex (separable) Hilbert space^{}. A von Neumann algebra $\mathcal{A}$ acting on $\mathscr{H}$ is a subset of the algebra of all bounded operators^{} $\mathcal{L}(\mathscr{H})$ such that:

(1)
$\mathcal{A}$ is closed under^{} the adjoint^{} operation (with the adjoint of an element $T$ denoted by ${T}^{*}$).

(2)
$\mathcal{A}$ equals its bicommutant, namely:
$$\mathcal{A}=\{A\in \mathcal{L}(\mathscr{H}):\forall B\in \mathcal{L}(\mathscr{H}),\forall C\in \mathcal{A},(BC=CB)\Rightarrow (AB=BA)\}.$$ (1.1)
If one calls a commutant of a set $\mathcal{A}$ the special set of bounded operators on $\mathcal{L}(\mathscr{H})$ which commute with all elements in $\mathcal{A}$, then this second condition implies that the commutant of the commutant of $\mathcal{A}$ is again the set $\mathcal{A}$.
On the other hand, a von Neumann algebra $\mathcal{A}$ inherits a unital subalgebra from $\mathcal{L}(\mathscr{H})$, and according to the first condition in its definition $\mathcal{A}$ 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 $\mathcal{A}$ is a von Neumann algebra if and only if $\mathrm{A}$ is a *subalgebra of $\mathrm{L}\mathit{}\mathrm{(}\mathrm{H}\mathrm{)}$, closed for the smallest topology^{} defined by continuous maps $\mathrm{(}\xi \mathrm{,}\eta \mathrm{)}\mathrm{\u27fc}\mathrm{(}A\mathit{}\xi \mathrm{,}\eta \mathrm{)}$ for all $$ where $$ denotes the inner product^{} defined on $\mathrm{H}$ . 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
$m$  $:A\otimes A\u27f6A,(multiplication)$  (1.2)  
$\eta $  $:\u2102\u27f6A,(unity)$ 
satisfying the conditions
$m(m\otimes \mathrm{\U0001d7cf})$  $=m(\mathrm{\U0001d7cf}\otimes m)$  (1.3)  
$m(\mathrm{\U0001d7cf}\otimes \eta )$  $=m(\eta \otimes \mathrm{\U0001d7cf})=\mathrm{id}.$ 
This first condition can be seen in terms of a commuting diagram :
$$\begin{array}{ccc}\hfill A\otimes A\otimes A\hfill & \hfill \stackrel{m\otimes \mathrm{id}}{\to}\hfill & \hfill A\otimes A\hfill \\ \hfill \mathrm{id}\otimes m\downarrow \hfill & & \hfill \downarrow m\hfill & & \\ \hfill A\otimes A\text{@}>m\gg A\hfill \end{array}$$  (1.4) 
Next suppose we consider ‘reversing the arrows’, and take an algebra $A$ equipped with a linear homorphisms $\mathrm{\Delta}:A\u27f6A\otimes A$, satisfying, for $a,b\in A$ :
$\mathrm{\Delta}(ab)$  $=\mathrm{\Delta}(a)\mathrm{\Delta}(b)$  (1.5)  
$(\mathrm{\Delta}\otimes \mathrm{id})\mathrm{\Delta}$  $=(\mathrm{id}\otimes \mathrm{\Delta})\mathrm{\Delta}.$ 
We call $\mathrm{\Delta}$ a comultiplication, which is said to be coasociative in so far that the following diagram commutes
$$  (1.6) 
There is also a counterpart to $\eta $, the counity map $\epsilon :A\u27f6\u2102$ satisfying
$$(\mathrm{id}\otimes \epsilon )\circ \mathrm{\Delta}=(\epsilon \otimes \mathrm{id})\circ \mathrm{\Delta}=\mathrm{id}.$$  (1.7) 
A bialgebra^{} $(A,m,\mathrm{\Delta},\eta ,\epsilon )$ is a linear space $A$ with maps $m,\mathrm{\Delta},\eta ,\epsilon $ satisfying the above properties.
Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism $S:A\u27f6A$, satisfying $S(ab)=S(b)S(a)$, for $a,b\in A$ . This map is defined implicitly via the property :
$$m(S\otimes \mathrm{id})\circ \mathrm{\Delta}=m(\mathrm{id}\otimes S)\circ \mathrm{\Delta}=\eta \circ \epsilon .$$  (1.8) 
We call $S$ the antipode map. A Hopf algebra is then a bialgebra $(A,m,\eta ,\mathrm{\Delta},\epsilon )$ 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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27
C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008)
arXiv:0709.4364v2 [quant–ph]
Title  weak Hopf algebra 
Canonical name  WeakHopfAlgebra 
Date of creation  20130322 18:12:43 
Last modified on  20130322 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 