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 $\epsilon$ is not necessarily a homomorphism of algebras.

• The axioms for the antipode map $S:A⟶A$ with respect to thecounit 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

$$\begin{array}{ccc}\hfill A\otimes A\hfill & \hfill \stackrel{S\otimes \mathrm{id}}{\to }\hfill & \hfill A\otimes A\hfill \\ \hfill \mathrm{\Delta }\uparrow \hfill & & \hfill \downarrow m\hfill & & \\ \hfill A>u\circ \epsilon \gg A\hfill \end{array}\mathit{\hspace{1em}\hspace{1em}}\begin{array}{ccc}\hfill A\otimes A\hfill & \hfill \stackrel{\mathrm{id}\otimes S}{\to }\hfill & \hfill A\otimes A\hfill \\ \hfill \mathrm{\Delta }\uparrow \hfill & & \hfill \downarrow m\hfill & & \\ \hfill A>u\circ \epsilon \gg A\hfill \end{array}$$(0.2)

along with the counit axiom:

(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 importantconcept in quantum operator algebra (QOA).

0.1 Examples of weak Hopf algebras

• We refer here to Bais et al. (2002). Let $G$ be a non-Abelian groupand $H\subset G$ a discrete subgroup. Let $F(H)$ denote the spaceof functions on $H$ and $ℂH$ the group algebra (which consistsof the linear span of group elements with the group structure).

  The quantum double $D(H)$ (Drinfeld, 1987) is defined by

  $$D(H)=F(H)\tilde{\otimes }ℂH,$$(0.4)

  where, for $x\in H$, the twisted tensor product is specified by

  $$\tilde{\otimes }\mapsto (f_1\otimes h_1)(f_2\otimes h_2)(x)=f_1(x)f_2(h_{1}x{h}_1^{-1})\otimes h_1{h}_2.$$(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

  $$ℛ\equiv \sigma \cdot ({\mathrm{\Pi }}_{\alpha }^A\otimes {\mathrm{\Pi }}_{\beta }^B)(R),$$(0.6)

  in terms of the Clebsch–Gordan series ${\mathrm{\Pi }}_{\alpha }^A\otimes {\mathrm{\Pi }}_{\beta }^B\cong N_{\alpha \beta C}^{AB\gamma }{\mathrm{\Pi }}_{\gamma }^C$, andwhere $\sigma$ denotes a flip operator. The operator ${ℛ}^2$ is sometimes called the monodromy orAharanov–Bohm phase factor. In the case of a condensate ina state $|v⟩$ in the carrier space of somerepresentation ${\mathrm{\Pi }}_{\alpha }^A$ . One considers the maximal Hopfsubalgebra $T$ of a Hopf algebra $A$ for which $|v⟩$is $T$–invariant; specifically  :

  $${\mathrm{\Pi }}_{\alpha }^A(P)|v⟩=\epsilon (P)|v⟩,\forall P\in T.$$(0.7)

• For the second example, consider $A=F(H)$ . The algebra offunctions 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 purelyelectric condensate $|v⟩$, the magnetic symmetryremains unbroken, but the electric symmetry $ℂH$ is broken to$ℂN_v$, with $N_v\subset H$, the stabilizer of $|v⟩$ . From this we obtain $T=F(H)\tilde{\otimes }ℂN_v$ .

• In Nikshych and Vainerman (2000) quantum groupoids (as weakC*–Hopf algebras, see below) were studied in relationship to thenoncommutative 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 $B_2=B_1⋊Q$ . Similarly, in Rehren (1997)‘paragroups’ (derived from weak C*–Hopf algebras) comprise(quantum) groupoids of equivalence classes such as associated with6j–symmetry groups (relative to a fusion rules algebra). Theycorrespond to type $II$ von Neumann algebras in quantum mechanics,and arise as symmetries where the local subfactors (in the senseof containment of observables within fields) have depth 2 in theJones extension. Related is how a von Neumann algebra $N$, such asof finite index depth 2, sits inside a weak Hopf algebra formed asthe crossed product $N⋊A$ (Böhm et al. 1999).

• In Mack and Schomerus (1992) using a more general notion of theDrinfeld construction, develop the notion of a quasitriangular quasi–Hopf algebra (QTQHA) is developed with the aimof studying a range of essential symmetries with specialproperties, 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 iscanonically associated with ${\mathrm{U}}_q({\mathrm{sl}}_2)$. Such QTQHAs areclaimed as the true symmetries of minimal conformal fieldtheories.

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 vonNeumann algebra $𝒜$ acting on $\mathscr{H}$ is a subset of the algebra ofall bounded operators $ℒ(\mathscr{H})$ such that:

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

• $𝒜$ equals its bicommutant, namely:

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

If one calls a commutant of a set $𝒜$ the special set ofbounded operators on $ℒ(\mathscr{H})$ which commute with all elements in$𝒜$, then this second condition implies that the commutant of thecommutant of $𝒜$ is again the set $𝒜$.

On the other hand, a von Neumann algebra $𝒜$ inherits aunital subalgebra from $ℒ(\mathscr{H})$, and according to thefirst condition in its definition $𝒜$ does indeed inherit a*-subalgebra structure, as further explained in the nextsection on C*-algebras. Furthermore, we have notableBicommutant Theorem which states that $𝒜$ is a vonNeumann 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 continuousmaps $\mathrm{(}\xi \mathrm{,}\eta \mathrm{)}\mathrm{⟼}\mathrm{(}A\mathit{}\xi \mathrm{,}\eta \mathrm{)}$ for all where denotes the inner product defined on $\mathrm{H}$ . Forfurther 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⟶A,(multiplication)$(1.2)

$\eta$$:ℂ⟶A,(unity)$

satisfying the conditions

$m(m\otimes \mathrm{𝟏})$$=m(\mathrm{𝟏}\otimes m)$(1.3)

$m(\mathrm{𝟏}\otimes \eta )$$=m(\eta \otimes \mathrm{𝟏})=\mathrm{id}.$

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

$$\begin{array}{ccc}\hfill A\otimes A\otimes A\hfill & \hfill \stackrel{m\otimes \mathrm{id}}{\to }\hfill & \hfill A\otimes A\hfill \\ \hfill \mathrm{id}\otimes m\downarrow \hfill & & \hfill \downarrow m\hfill & & \\ \hfill A\otimes A>m\gg A\hfill \end{array}$$(1.4)

Next suppose we consider ‘reversing the arrows’, and take analgebra $A$ equipped with a linear homorphisms $\mathrm{\Delta }:A⟶A\otimes A$, 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 becoasociative in so far that the following diagram commutes

(1.6)

There is also a counterpart to $\eta$, the counity map$\epsilon :A⟶ℂ$ 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 mustappend such a bialgebra with an antihomomorphism $S:A⟶A$,satisfying $S(ab)=S(b)S(a)$, for $a,b\in A$ . This map isdefined 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 thena bialgebra $(A,m,\eta ,\mathrm{\Delta },\epsilon )$ equipped with an antipodemap $S$ .

Commutative and noncommutative Hopf algebras form the backbone ofquantum ‘groups’ and are essential to the generalizations ofsymmetry. Indeed, in most respects a quantum ‘group’ is identifiablewith a Hopf algebra. When such algebras are actuallyassociated with proper groups of matrices there isconsiderable scope for their representations on both finiteand infinite dimensional Hilbert spaces.