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# 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 $\varepsilon$ is not necessarily a homomorphism of algebras.

- (3)
The axioms for the antipode map $S:A{\longrightarrow}A$ with respect to the counit are as follows. For all $h\in H$,

$\displaystyle m({\rm id}\otimes S)\Delta(h)$ $\displaystyle=(\varepsilon\otimes{\rm id})(\Delta(1)(h\otimes 1))$ (0.1) $\displaystyle m(S\otimes{\rm id})\Delta(h)$ $\displaystyle=({\rm id}\otimes\varepsilon)((1\otimes h)\Delta(1))$ $\displaystyle S(h)$ $\displaystyle=S(h_{{(1)}})h_{{(2)}}S(h_{{(3)}})~{}.$

These axioms may be appended by the following commutative diagrams

${\begin{matrix}A\otimes A&\cd@stack{\rightarrowfill@}{S\otimes{\rm id}}{}&A% \otimes A\\ {\Delta}{\Big\uparrow}&&{}{\Big\downarrow}{m}&&\\ A@ >u\circ\varepsilon>>A\end{matrix}}\qquad{\begin{matrix}A\otimes A&\cd@stack% {\rightarrowfill@}{{\rm id}\otimes S}{}&A\otimes A\\ {\Delta}{\Big\uparrow}&&{}{\Big\downarrow}{m}&&\\ A@ >u\circ\varepsilon>>A\end{matrix}}$ | (0.2) |

along with the counit axiom:

$\xymatrix@C=3pc@R=3pc{A\otimes A\ar[d]_{{\varepsilon\otimes 1}}&A\ar[l]_{{% \Delta}}\ar[dl]_{{{\rm id}_{A}}}\ar[d]^{{\Delta}}\\ A&A\otimes A\ar[l]^{{1\otimes\varepsilon}}}$ | (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 non-Abelian group and $H\subset G$ a discrete subgroup. Let $F(H)$ denote the space of functions on $H$ and $\mathbb{C}H$ 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$D(H)=F(H)~{}\widetilde{\otimes}~{}\mathbb{C}H~{},$ (0.4) where, for $x\in H$, the twisted tensor product is specified by

$\widetilde{\otimes}\mapsto~{}(f_{1}\otimes h_{1})(f_{2}\otimes h_{2})(x)=f_{1}% (x)f_{2}(h_{1}xh_{1}^{{-1}})\otimes h_{1}h_{2}~{}.$ (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 $\varepsilon$, the double $D(H)$ has a trivial representation given by $\varepsilon(f\otimes h)=f(e)$ . 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

$\mathcal{R}\equiv\sigma\cdot(\Pi^{A}_{{\alpha}}\otimes\Pi^{B}_{{\beta}})(R)~{},$ (0.6) in terms of the Clebsch–Gordan series $\Pi^{A}_{{\alpha}}\otimes\Pi^{B}_{{\beta}}\cong N^{{AB\gamma}}_{{\alpha\beta C% }}~{}\Pi^{C}_{{\gamma}}$, 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 $|v\rangle$ in the carrier space of some representation $\Pi^{A}_{{\alpha}}$ . One considers the maximal Hopf subalgebra $T$ of a Hopf algebra $A$ for which $|v\rangle$ is $T$–invariant; specifically :$\Pi^{A}_{{\alpha}}(P)~{}|v\rangle=\varepsilon(P)|v\rangle~{},~{}\forall P\in T% ~{}.$ (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, $HKH^{{-1}}=K$ . Next, consider $A=D(H)$ . On breaking a purely electric condensate $|v\rangle$, the magnetic symmetry remains unbroken, but the electric symmetry $\mathbb{C}H$ is broken to $\mathbb{C}N_{v}$, with $N_{v}\subset H$, the stabilizer of $|v\rangle$ . From this we obtain $T=F(H)\widetilde{\otimes}\mathbb{C}N_{v}$ .

- (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

$A\subset B\subset B_{1}\subset B_{2}\subset\ldots$ (0.8) is the Jones extension induced by a finite index depth $2$ inclusion $A\subset B$ of $II_{1}$ 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 $B_{2}=B_{1}\rtimes Q$ . 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 $II$ 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 $N\rtimes A$ (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 ${\rm U}_{q}(\rm{sl}_{2})$ with $|q|=1$ . If $q^{p}=1$, then it is shown that a QTQHA is canonically associated with ${\rm U}_{q}(\rm{sl}_{2})$. 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 $\mathcal{H}$ denote a complex (separable) Hilbert space. A *von
Neumann algebra* $\mathcal{A}$ acting on $\mathcal{H}$ is a subset of the algebra of
all bounded operators $\mathcal{L}(\mathcal{H})$ 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:

$\mathcal{A}=\{A\in\mathcal{L}(\mathcal{H}):\forall B\in\mathcal{L}(\mathcal{H}% ),\forall C\in\mathcal{A},~{}(BC=CB)\Rightarrow(AB=BA)\}~{}.$ (1.1)

If one calls a *commutant* of a set $\mathcal{A}$ the special set of
bounded operators on $\mathcal{L}(\mathcal{H})$ 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 $\mathcal{L}(\mathcal{H})$, 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 $\mathcal{A}$ is a *-subalgebra of
$\mathcal{L}(\mathcal{H})$, closed for the smallest topology defined by continuous
maps $(\xi,\eta)\longmapsto(A\xi,\eta)$ for all $<A\xi,\eta)>$
where $<.,.>$ denotes the inner product defined on $\mathcal{H}$* . 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

$\displaystyle m$ | $\displaystyle:A\otimes A{\longrightarrow}A~{},~{}(multiplication)$ | (1.2) | ||

$\displaystyle\eta$ | $\displaystyle:\mathbb{C}{\longrightarrow}A~{},~{}(unity)$ |

satisfying the conditions

$\displaystyle m(m\otimes\mathbf{1})$ | $\displaystyle=m(\mathbf{1}\otimes m)$ | (1.3) | ||

$\displaystyle m(\mathbf{1}\otimes\eta)$ | $\displaystyle=m(\eta\otimes\mathbf{1})={\rm id}~{}.$ |

This first condition can be seen in terms of a commuting diagram :

$\begin{matrix}A\otimes A\otimes A&\cd@stack{\rightarrowfill@}{m\otimes{\rm id}% }{}&A\otimes A\\ {{\rm id}\otimes m}{\Big\downarrow}&&{}{\Big\downarrow}{m}&&\\ A\otimes A@ >m>>A\end{matrix}$ | (1.4) |

Next suppose we consider ‘reversing the arrows’, and take an algebra $A$ equipped with a linear homorphisms $\Delta:A{\longrightarrow}A\otimes A$, satisfying, for $a,b\in A$ :

$\displaystyle\Delta(ab)$ | $\displaystyle=\Delta(a)\Delta(b)$ | (1.5) | ||

$\displaystyle(\Delta\otimes{\rm id})\Delta$ | $\displaystyle=({\rm id}\otimes\Delta)\Delta~{}.$ |

We call $\Delta$ a *comultiplication*, which is said to be
*coasociative* in so far that the following diagram commutes

$\begin{matrix}A\otimes A\otimes A&\cd@stack{\leftarrowfill@}{\Delta\otimes{\rm id% }}{}&A\otimes A\\ {{\rm id}\otimes\Delta}{\Big\uparrow}&&{}{\Big\uparrow}{\Delta}&&\\ A\otimes A@ <\Delta<<A\end{matrix}$ | (1.6) |

There is also a counterpart to $\eta$, the *counity* map
$\varepsilon:A{\longrightarrow}\mathbb{C}$ satisfying

$({\rm id}\otimes\varepsilon)\circ\Delta=(\varepsilon\otimes{\rm id})\circ% \Delta={\rm id}~{}.$ | (1.7) |

A *bialgebra* $(A,m,\Delta,\eta,\varepsilon)$ is a linear space $A$ with maps $m,\Delta,\eta,\varepsilon$
satisfying the above properties.

Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism $S:A{\longrightarrow}A$, satisfying $S(ab)=S(b)S(a)$, for $a,b\in A$ . This map is defined implicitly via the property :

$m(S\otimes{\rm id})\circ\Delta=m({\rm id}\otimes S)\circ\Delta=\eta\circ% \varepsilon~{}~{}.$ | (1.8) |

We call $S$ the *antipode map*. A *Hopf algebra* is then
a bialgebra $(A,m,\eta,\Delta,\varepsilon)$ 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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arXiv:0709.4364v2 [quant–ph]

## Mathematics Subject Classification

08C99*no label found*81R15

*no label found*57T05

*no label found*81R50

*no label found*16W30

*no label found*

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