PlanetPhysics/Weak Hopf C Algebra 2

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A weak Hopf $$C^*$$-algebra is defined as a weak Hopf algebra 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. 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 weakening the definining axioms of a Hopf algebra as follows~:

\item[(1)] The comultiplication is not necessarily unit-preserving. \item[(2)] The counit $$\vep$$ is not necessarily a homomorphism of algebras. \item[(3)] The axioms for the antipode map $$S : A \lra A$$ with respect to the counit are as follows. For all $$h \in H$$, "$ m(\ID \otimes S) \Delta (h) &= (\vep \otimes \ID)(\Delta (1) (h \otimes 1)) \\ m(S \otimes \ID) \Delta (h) &= (\ID \otimes \vep)((1 \otimes h) \Delta(1)) \\ S(h) &= S(h_{(1)}) h_{(2)} S(h_{(3)}) ~. $"

These axioms may be appended by the following commutative diagrams $$ {\begin{CD} A \otimes A @> S\otimes \ID >> A \otimes A \\ @A \Delta AA @VV m V \\ A @ > u \circ \vep >> A \end{CD}} \qquad {\begin{CD} A \otimes A @> \ID\otimes S >> A \otimes A \\ @A \Delta AA @VV m V \\ A @ > u \circ \vep >> A \end{CD}} $$ along with the counit axiom: $$ \xymatrix@C=3pc@R=3pc{ A \otimes A \ar[d]_{\vep \otimes 1} & A \ar[l]_{\Delta} \ar[dl]_{\ID_A} \ar[d]^{\Delta} \\ A & A \otimes A \ar[l]^{1 \otimes \vep}} $$

Some authors substitute the term quantum `groupoid' for a weak Hopf algebra.

Examples of weak Hopf C*-algebra.
\item[(1)] In Nikshych and Vainerman (2000) quantum groupoids were considered as weak C*--Hopf algebras and 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 $" is the Jones extension induced by a finite index depth $$2$$ inclusion $$A \subset B$$ of $$II_1$$ factors, then $$Q= A' \cap B_2$$ 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 $$N \rtimes A$$ (B\"ohm et al. 1999). \item[(2)] In Mack and Schomerus (1992) using a more general notion of the Drinfeld construction, develop the notion of a \emph{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 $$\U_q (\rm{sl}_2)$$ with $$\vert q \vert =1$$~. If $$q^p=1$$, then it is shown that a QTQHA is canonically associated with $$\U_q (\rm{sl}_2)$$. Such QTQHAs are claimed as the true symmetries of minimal conformal field theories.

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

Let $$\mathbb{H}$$ denote a complex (separable) Hilbert space. A \emph{von Neumann algebra} $$\A$$ acting on $$\mathbb{H}$$ is a subset of the $$*$$--algebra of all bounded operators $$\cL(\mathbb{H})$$ such that:

\item[(1)] $$\A$$ is closed under the adjoint operation (with the adjoint of an element $$T$$ denoted by $$T^*$$). \item[(2)] $$\A$$ equals its bicommutant, namely: "$ \A= \{A \in \cL(\mathbb{H}) : \forall B \in \cL(\mathbb{H}), \forall C\in \A,~ (BC=CB)\Rightarrow (AB=BA)\}~. $"

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

On the other hand, a von Neumann algebra $$\A$$ inherits a unital subalgebra from $$\cL(\mathbb{H})$$, and according to the first condition in its definition $$\A$$ does indeed inherit a *-subalgebra structure, as further explained in the next section on C*-algebras. Furthermore, we have the notable Bicommutant theorem which states that $$\A$$ \emph{is a von Neumann algebra if and only if $$\A$$ is a *-subalgebra of $$\cL(\mathbb{H})$$, closed for the smallest topology defined by continuous maps $$(\xi,\eta)\longmapsto (A\xi,\eta)$$ for all $$$$ where $$<.,.>$$ denotes the inner product defined on $$\mathbb{H}$$}~. For further instruction on this subject, see e.g. Aflsen and Schultz (2003), Connes (1994). \\

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.