weak Hopf C*-algebra
A weak Hopf -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 .
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:
These axioms may be appended by the following commutative diagrams
along with the counit axiom:
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 weak C*–Hopf algebras and were studied in relationship to the noncommutative symmetries of depth 2 von Neumann subfactors. If
is the Jones extension induced by a finite index depth inclusion of factors, then admits a quantum groupoid structure and acts on , so that and . 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 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 in the Jones extension. Related is how a von Neumann algebra , such as of finite index depth , sits inside a weak Hopf algebra formed as the crossed product (Böhm et al. 1999).
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 with . If , then it is shown that a QTQHA is canonically associated with . Such QTQHAs are claimed as the true symmetries of minimal conformal field theories.
0.2 Von Neumann Algebras (or -algebras).
equals its bicommutant, namely:
If one calls a commutant of a set the special set of bounded operators on which commute with all elements in , then this second condition implies that the commutant of the commutant of is again the set .
On the other hand, a von Neumann algebra inherits a unital subalgebra from , and according to the first condition in its definition 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 is a von Neumann algebra if and only if is a *-subalgebra of , closed for the smallest topology defined by continuous maps for all where denotes the inner product defined on . 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.
- 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.
B. Fauser: A treatise on quantum Clifford Algebras. Konstanz, Habilitationsschrift.
- 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, Editor. 2003. https://perswww.kuleuven.be/ u0018768/artikels/strasbourg.pdfLocally 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.
- 25 http://planetmath.org/?op=getobj&from=books&id=294Stefaan Vaes and Leonid Vainerman.2003. On Low-Dimensional Locally Compact Quantum Groups in Locally Compact Quantum Groups and Groupoids: Proceedings of the Meeting of Theoretical Physicists and Mathematicians
|Title||weak Hopf C*-algebra|
|Date of creation||2013-03-22 18:12:47|
|Last modified on||2013-03-22 18:12:47|
|Last modified by||bci1 (20947)|
|Defines||weak Hopf algebra|
|Defines||weak Hopf C*-algebra|
|Defines||von Neumann algebra|