weak Hopf C*-algebra

Definition 0.1.

A weak Hopf $C^{\mathrm{*}}$-algebra is defined as a weak Hopf algebra (http://planetmath.org/WeakHopfCAlgebra) which admits a faithful $*$–representation on a Hilbert space. The weak C*–Hopf algebra is therefore much more likely to be closely related to a quantum groupoid than the weak Hopf algebra. However, one can argue that locally compact groupoids equipped with a Haar measure are even closer to defining quantum groupoids (http://planetmath.org/QuantumGroupoids2).

There are already several, significant examples that motivate the consideration of weak C*-Hopf algebras which also deserve mentioning in the context of standard quantum theories. Furthermore, notions such as (proper) weak C*-algebroids can provide the main framework for symmetry breaking and quantum gravity that we are considering here. Thus, one may consider the quasi-group symmetries constructed by means of special transformations of the coordinate space $M$.

Remark:Recall that the weak Hopf algebra is defined as the extension of a Hopf algebra by weakeningthe definining 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.

0.1 Examples of weak Hopf C*-algebra.

• In Nikshych and Vainerman (2000) quantum groupoids were considered as weakC*–Hopf algebras and were studied in relationship to thenoncommutative symmetries of depth 2 von Neumann subfactors. If

  $$A\subset B\subset B_1\subset B_2\subset \mathrm{\dots }$$(0.4)

  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.

0.2 Von Neumann Algebras (or $W^{*}$-algebras).

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:

  $$𝒜=\{A\in ℒ(\mathscr{H}):\forall B\in ℒ(\mathscr{H}),\forall C\in 𝒜,(BC=CB)\Rightarrow (AB=BA)\}.$$(0.5)

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 the 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).

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.