PlanetPhysics/Groupoid and Group Representations

From Wikiversity
Jump to navigation Jump to search

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

Groupoid representations[edit | edit source]

Whereas group representations of quantum unitary operators are extensively employed in standard quantum mechanics, the applications of groupoid representations are still under development. For example, a description of stochastic quantum mechanics in curved spacetime (Drechsler and Tuckey, 1996) involving a Hilbert bundle is possible in terms of groupoid representations which can indeed be defined on such a Hilbert bundle , but cannot be expressed as the simpler group representations on a Hilbert space . On the other hand, as in the case of group representations, unitary groupoid representations induce associated C*-algebra representations. In the next subsection we recall some of the basic results concerning groupoid representations and their associated groupoid *-algebra representations. For further details and recent results in the mathematical theory of groupoid representations one has also available the succint monograph by Buneci (2003) and references cited therein ( ).

Let us consider first the relationships between these mainly algebraic concepts and their extended quantum symmetries, also including relevant computation examples. Let us consider first several further extensions of symmetry and algebraic topology in the context of local quantum physics/ quantum field theory, symmetry breaking, quantum chromodynamics and the development of novel supersymmetry theories of quantum gravity. In this respect one can also take spacetime 'inhomogeneity' as a criterion for the comparisons between physical, partial or local, symmetries: on the one hand, the example of paracrystals reveals Thermodynamic disorder (entropy) within its own spacetime framework, whereas in spacetime itself, whatever the selected model, the inhomogeneity arises through (super) gravitational effects. More specifically, in the former case one has the technique of the generalized Fourier--Stieltjes transform (along with convolution and Haar measure), and in view of the latter, we may compare the resulting 'broken'/paracrystal--type symmetry with that of the supersymmetry predictions for weak gravitational fields (e.g., 'ghost' particles) along with the broken supersymmetry in the presence of intense gravitational fields. Another significant extension of quantum symmetries may result from the superoperator algebra/algebroids of Prigogine's quantum superoperators which are defined only for irreversible, infinite-dimensional systems (Prigogine, 1980).

Definition of extended quantum groupoid and algebroid symmetries[edit | edit source]

Quantum groups~ Representations ~ weak Hopf algebras ~ ~quantum groupoids and algebroids Our intention here is to view the latter scheme in terms of weak Hopf C*--algebroid-- and/or other-- extended symmetries, which we propose to do, for example, by incorporating the concepts of rigged Hilbert spaces and \emph{sectional functions for a small category}. We note, however, that an alternative approach to quantum 'groupoids' has already been reported (Maltsiniotis, 1992), (perhaps also related to noncommutative geometry); this was later expressed in terms of deformation-quantization: the Hopf algebroid deformation of the universal enveloping algebras of Lie algebroids (Xu, 1997) as the classical limit of a quantum 'groupoid'; this also parallels the introduction of quantum 'groups' as the deformation-quantization of Lie bialgebras. Furthermore, such a Hopf algebroid approach (Lu, 1996) leads to categories of Hopf algebroid modules (Xu, 1997) which are monoidal, whereas the links between Hopf algebroids and monoidal bicategories were investigated by Day and Street (1997).

As defined under the following heading on groupoids, let be a locally compact groupoid endowed with a (left) Haar system, and let be the convolution --algebra (we append with if necessary, so that is unital). Then consider such a groupoid representation

that respects a compatible measure on (cf Buneci, 2003). On taking a state on , we assume a parametrization Furthermore, each is considered as a \emph{rigged Hilbert space} Bohm and Gadella (1989), that is, one also has the following nested inclusions: in the usual manner, where is a dense subspace of with the appropriate locally convex topology, and is the space of continuous antilinear functionals of ~. For each , we require to be invariant under and is a continuous representation of on ~. With these conditions, representations of (proper) quantum groupoids that are derived for weak C*--Hopf algebras (or algebroids) modeled on rigged Hilbert spaces could be suitable generalizations in the framework of a Hamiltonian generated semigroup of time evolution of a quantum system via integration of Schr\"odinger's equation as studied in the case of Lie groups (Wickramasekara and Bohm, 2006). The adoption of the rigged Hilbert spaces is also based on how the latter are recognized as reconciling the Dirac and von Neumann approaches to quantum theories (Bohm and Gadella, 1989).

Next, let be a locally compact Hausdorff groupoid and a locally compact Hausdorff space. ( will be called a locally compact groupoid, or lc- groupoid for short). In order to achieve a small C*--category we follow a suggestion of A. Seda (private communication) by using a general principle in the context of Banach bundles (Seda, 1976, 982)). Let be a continuous, open and surjective map. For each , consider the fibre , and set equipped with a uniform norm ~. Then we set ~. We form a Banach bundle as follows. Firstly, the projection is defined via the typical fibre ~. Let denote the continuous complex valued functions on with compact support. We obtain a sectional function defined via restriction as ~. Commencing from the vector space , the set is dense in ~. For each , the function is continuous on , and each is a continuous section of ~. These facts follow from Seda (1982, theorem 1). Furthermore, under the convolution product , \textit{the space forms an associative algebra over Failed to parse (unknown function "\bC"): {\displaystyle \bC} } (cf. Seda, 1982, Theorem 3).

Groupoids[edit | edit source]

Recall that a groupoid is, loosely speaking, a small category with inverses over its set of objects ~. One often writes for the set of morphisms in from to ~. A topological groupoid consists of a space , a distinguished subspace , called {\it the space of objects} of , together with maps Failed to parse (unknown function "\xymatrix"): {\displaystyle r,s~:~ \xymatrix{{\mathsf{G}} \ar@<1ex>[r]^r \ar[r]_s & {\mathsf{G}}^{(0)}} } called the {\it range} and {\it source maps} respectively, together with a law of composition such that the following hold~:~

\item[(1)] ~, for all ~. \item[(2)] ~, for all ~. \item[(3)] ~, for all ~. \item[(4)] ~. \item[(5)] Each has a two--sided inverse with ~.

Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call the set of objects of ~. For , the set of arrows forms a group , called the isotropy group of at . Thus, as is well kown, a topological groupoid is just a groupoid internal to the category of topological spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalise bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to Brown (2006).

Several examples of groupoids are:

  • (a) locally compact groups, transformation groups, and any group in general (e.g. [59]
  • (b) equivalence relations
  • (c) tangent bundles
  • (d) the tangent groupoid (e.g. [4])
  • (e) holonomy groupoids for foliations (e.g. [4])
  • (f) Poisson groupoids (e.g. [81])
  • (g) graph groupoids (e.g. [47, 64]).

As a simple example of a groupoid, consider (b) above. Thus, let R be an equivalence relation on a set X. Then R is a groupoid under the following operations: . Here, , (the diagonal of ) and .

Thus, = . When , R is called a trivial groupoid. A special case of a trivial groupoid is . (So every i is equivalent to every j ). Identify with the matrix unit . Then the groupoid is just matrix multiplication except that we only multiply when , and . We do not really lose anything by restricting the multiplication, since the pairs excluded from groupoid multiplication just give the 0 product in normal algebra anyway.

For a groupoid to be a locally compact groupoid means that is required to be a (second countable) locally compact Hausdorff space , and the product and also inversion maps are required to be continuous. Each as well as the unit space is closed in .

What replaces the left Haar measure on is a system of measures (), where is a positive regular Borel measure on with dense support. In addition, the 's are required to vary continuously (when integrated against and to form an invariant family in the sense that for each x, the map is a measure preserving homeomorphism from onto . Such a system is called a left Haar system for the locally compact groupoid .

This is defined more precisely next.

Haar systems for locally compact topological groupoids[edit | edit source]

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \xymatrix{ {\mathsf{G}} \ar@<1ex>[r]^r \ar[r]_s & {\mathsf{G}}^{(0)}}=X } be a locally compact, locally trivial topological groupoid with its transposition into transitive (connected) components. Recall that for , the costar of denoted is defined as the closed set , whereby is a principal --bundle relative to fixed base points ~. Assuming all relevant sets are locally compact, then following Seda (1976), a (left) Haar system on denoted (for later purposes), is defined to comprise of i) a measure on , ii) a measure on and iii) a measure on such that for every Baire set of , the following hold on setting ~:

 \item[(1)]  is measurable. \item[(2)]  ~. \item[(3)] , for all  and ~.

The presence of a left Haar system on has important topological implications: it requires that the range map is open. For such a with a left Haar system, the vector space is a convolution *--algebra , where for :

with .

One has to be the enveloping C*--algebra of (and also representations are required to be continuous in the inductive limit topology). Equivalently, it is the completion of where is the universal representation of . For example, if , then is just the finite dimensional algebra , the span of the s.

There exists (cf. [1]) a measurable Hilbert bundle with and a G-representation L on . Then, for every pair of square integrable sections of , it is required that the function be --measurable. The representation of is then given by:\\ .

The triple is called a measurable --Hilbert bundle.

All Sources[edit | edit source]

[2] [3] [4] [5] [6] [7] [1] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

References[edit | edit source]

  1. 1.0 1.1 M. R. Buneci.: Groupoid Representations , (orig. title "Reprezentari de Grupoizi"), Ed. Mirton: Timishoara (2003).
  2. E. M. Alfsen and F. W. Schultz: Geometry of State Spaces of Operator Algebras , Birkhäuser, Boston--Basel--Berlin (2003).
  3. I. C. 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).
  4. 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).
  5. I.C.Baianu, R. Brown J.F. Glazebrook, and G. Georgescu, Towards Quantum Non--Abelian Algebraic Topology. in preparation , (2008).
  6. 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).
  7. J.W. Barrett.: Geometrical measurements in three-dimensional quantum gravity. Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics (2001). Intl. J. Modern Phys. A 18 , October, suppl., 97--113 (2003).
  8. M. Chaician and A. Demichev: Introduction to Quantum Groups , World Scientific (1996).
  9. Coleman and De Luccia: Gravitational effects on and of vacuum decay., Phys. Rev. D 21 : 3305 (1980).
  10. 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).
  11. W. Drechsler and P. A. Tuckey: On quantum and parallel transport in a Hilbert bundle over spacetime., Classical and Quantum Gravity , 13 :611-632 (1996). doi: 10.1088/0264--9381/13/4/004
  12. V. G. Drinfel'd: Quantum groups, In Proc. Intl. Congress of Mathematicians, Berkeley 1986, (ed. A. Gleason), Berkeley, 798-820 (1987).
  13. G. J. Ellis: Higher dimensional crossed modules of algebras, J. of Pure Appl. Algebra 52 (1988), 277-282.
  14. 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).
  15. P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19-52 (1999)
  16. 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.
  17. B. Fauser: A treatise on quantum Clifford Algebras . Konstanz, Habilitationsschrift. arXiv.math.QA/0202059 (2002).
  18. 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).
  19. J. M. G. Fell.: The Dual Spaces of C*--Algebras., Transactions of the American Mathematical Society, 94 : 365--403 (1960).
  20. F.M. Fernandez and E. A. Castro.: (Lie) Algebraic Methods in Quantum Chemistry and Physics. , Boca Raton: CRC Press, Inc (1996).
  21. 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).]
  22. A.~Fröhlich: Non--Abelian Homological Algebra. {I}. {D}erived functors and satellites., Proc. London Math. Soc. , 11 (3): 239--252 (1961).
  23. R. Gilmore: Lie Groups, Lie Algebras and Some of Their Applications. , Dover Publs., Inc.: Mineola and New York, 2005.
  24. P. Hahn: Haar measure for measure groupoids., Trans. Amer. Math. Soc . 242 : 1--33(1978).
  25. P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc . 242 :34--72(1978).
  26. R. Heynman and S. Lifschitz. 1958. Lie Groups and Lie Algebras ., New York and London: Nelson Press.
  27. C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008) arXiv:0709.4364v2 [quant--ph].