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weak Hopf C*-algebra (Definition)
Definition 0.1   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 :

(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 $h \in H$,
\begin{equation*}\begin{aligned}m({\rm id}\otimes S) \Delta (h) &= (\varepsilon ... ...lta(1)) \\ S(h) &= S(h_{(1)}) h_{(2)} S(h_{(3)}) ~. \end{aligned}\end{equation*}

These axioms may be appended by the following commutative diagrams

$\displaystyle {\begin{CD}A \otimes A @> S\otimes {\rm id}>> A \otimes A \\ @A \... ...A \otimes A \\ @A \Delta AA @VV m V \\ A @ > u \circ \varepsilon >> A \end{CD}}$ (0.2)
along with the counit axiom:
$\displaystyle \xymatrix@C=3pc@R=3pc{ A \otimes A \ar[d]_{\varepsilon \otimes 1}... ...{\rm id}_A} \ar[d]^{\Delta} \\ A & A \otimes A \ar[l]^{1 \otimes \varepsilon }}$ (0.3)

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

Examples of weak Hopf C*-algebra.

(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

$\displaystyle A \subset B \subset B_1 \subset B_2 \subset \ldots$ (0.4)
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 $N \rtimes A$ (Böhm et al. 1999).
(2)  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 $\vert q \vert =1$ . 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.

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

Let $\H$ denote a complex (separable) Hilbert space. A von Neumann algebra $\mathcal A$ acting on $\H$ is a subset of the $*$–algebra of all bounded operators $\mathcal L(\H )$ 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:
$\displaystyle \mathcal A= \{A \in \mathcal L(\H ) : \forall B \in \mathcal L(\H ), \forall C\in \mathcal A,~ (BC=CB)\Rightarrow (AB=BA)\}~.$ (0.5)

If one calls a commutant of a set $\mathcal A$ the special set of bounded operators on $\mathcal L(\H )$ 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 $\mathcal L(\H )$, 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 the notable Bicommutant theorem which states that $\mathcal A$ is a von Neumann algebra if and only if $\mathcal A$ is a *-subalgebra of $\mathcal L(\H )$, 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 $\H$ . 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.

Bibliography

1
E. M. Alfsen and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkhäuser, Boston–Basel–Berlin (2003).
2
I. Baianu : Categories, Functors and Automata Theory: A Novel Approach to Quantum Automata through Algebraic–Topological Quantum Computations., Proceed. 4th Intl. Congress LMPS, (August-Sept. 1971).
3
I. C. Baianu, J. F. Glazebrook and R. Brown.: A Non–Abelian, Categorical Ontology of Spacetimes and Quantum Gravity., Axiomathes 17,(3-4): 353-408(2007).
4
I.C.Baianu, R. Brown J.F. Glazebrook, and G. Georgescu, Towards Quantum Non–Abelian Algebraic Topology. in preparation, (2008).
5
F.A. Bais, B. J. Schroers and J. K. Slingerland: Broken quantum symmetry and confinement phases in planar physics, Phys. Rev. Lett. 89 No. 18 (1–4): 181–201 (2002).
6
M. R. Buneci.: Groupoid Representations, Ed. Mirton: Timishoara (2003).
7
M. Chaician and A. Demichev: Introduction to Quantum Groups, World Scientific (1996).
8
L. Crane and I.B. Frenkel. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. Topology and physics. J. Math. Phys. 35 (no. 10): 5136–5154 (1994).
9
V. G. Drinfel'd: Quantum groups, In Proc. Intl. Congress of Mathematicians, Berkeley 1986, (ed. A. Gleason), Berkeley, 798-820 (1987).
10
G. J. Ellis: Higher dimensional crossed modules of algebras, J. of Pure Appl. Algebra 52 (1988), 277-282.
11
P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups, Comm.Math.Phys., 196: 591-640 (1998).
12
P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19-52 (1999)
13
P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang–Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999), pp. 89-129, Cambridge University Press, Cambridge, 2001.
14
B. Fauser: A treatise on quantum Clifford Algebras. Konstanz, Habilitationsschrift.
arXiv.math.QA/0202059 (2002).
15
B. Fauser: Grade Free product Formulae from Grassman–Hopf Gebras. Ch. 18 in R. Ablamowicz, Ed., Clifford Algebras: Applications to Mathematics, Physics and Engineering, Birkhäuser: Boston, Basel and Berlin, (2004).
16
J. M. G. Fell.: The Dual Spaces of C*–Algebras., Transactions of the American Mathematical Society, 94: 365–403 (1960).
17
F.M. Fernandez and E. A. Castro.: (Lie) Algebraic Methods in Quantum Chemistry and Physics., Boca Raton: CRC Press, Inc (1996).
18
R. P. Feynman: Space–Time Approach to Non–Relativistic Quantum Mechanics, Reviews of Modern Physics, 20: 367–387 (1948). [It is also reprinted in (Schwinger 1958).]
19
A. Fröhlich: Non–Abelian Homological Algebra. I. Derived functors and satellites., Proc. London Math. Soc., 11(3): 239–252 (1961).
20
R. Gilmore: Lie Groups, Lie Algebras and Some of Their Applications., Dover Publs., Inc.: Mineola and New York, 2005.
21
P. Hahn: Haar measure for measure groupoids., Trans. Amer. Math. Soc. 242: 1–33(1978).
22
P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc. 242:34–72(1978).
23
R. Heynman and S. Lifschitz. 1958. Lie Groups and Lie Algebras., New York and London: Nelson Press.
24
Leonid Vainerman. 2003. Locally Compact Quantum Groups and Groupoids: Proceedings of the Meeting of Theoretical Physicists and Mathematicians., Strasbourg, February 21-23, 2002., Walter de Gruyter Gmbh & Co: Berlin.



"weak Hopf C*-algebra" is owned by bci1.

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Other names:  quantum groupoid dual
Also defines:  commutant, von Neumann algebra, $*$--representation on a Hilbert space, weak C*-algebroid, coordinate space, unital subalgebra, bicommutant
Keywords:  commutant, von Neumann algebra, C*-algebra, Haar measure associated with a groupoid, bicommutant theorem

Cross-references: representations, matrices, inner product, theorem, C*-algebras, section, commute, operation, operators, quantum group, fields, observables, quantum mechanics, type, groups, groupoids, paragroups, noncommutative, commutative diagrams, homomorphism, Hopf algebra, quantum gravity, quantum theories, Haar measure, locally compact groupoids, quantum groupoid, Hilbert space
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This is version 2 of weak Hopf C*-algebra, born on 2009-05-12, modified 2009-06-22.
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 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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