PlanetPhysics/Quantum Groupoids

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This is a topic entry on quantum groupoids, related mathematical concepts and their applications in modern quantum phyiscs.
quantum groupoids , Failed to parse (unknown function "\grp"): {\displaystyle Q_{\grp}}
' s, are currently defined either as quantized, locally compact groupoids endowed with a left Haar measure system, Failed to parse (unknown function "\grp"): {\displaystyle (\grp,\mu)}
, or as weak Hopf algebras (WHA). This concept is also an extension of the notion of quantum group, which is sometimes represented by a Hopf algebra, . Quantum groupoid representations define extended quantum symmetries beyond the `Standard Model' (SUSY) in mathematical physics or noncommutative geometry.

Mathematical Definitions and Related Physical ExplanationsEdit

Quantum Groupoids and the Groupoid C*--AlgebraEdit

Quantum groupoid (e.g., weak Hopf algebras) and algebroid symmetries figure prominently both in the theory of dynamical deformations of quantum `groups' (e.g., Hopf algebras) and the quantum Yang--Baxter equations (Etingof et al., 1999,2001). On the other hand, one can also consider the natural extension of locally compact (quantum) groups to locally compact (proper) groupoids equipped with a Haar measure and a corresponding groupoid representation theory (Buneci, 2003) as a major, potentially interesting source for locally compact (but generally non-Abelian) quantum groupoids . The corresponding quantum groupoid representations on bundles of Hilbert spaces extend quantum symmetries well beyond those of quantum `groups'/Hopf algebras and simpler operator algebra representations, and are also consistent with the locally compact quantum group representations that were recently studied in some detail by Kustermans and Vaes (2000, and references cited therein). The latter quantum groups are neither Hopf algebras, nor are they equivalent to Hopf algebras or their dual coalgebras. Quantum groupoid representations are, however, the next important step towards unifying quantum field theories with general relativity in a locally covariant and quantized form. Such representations need not however be restricted to weak Hopf algebra representations, as the latter have no known connection to any type of GR theory and also appear to be inconsistent with GR.

One is also motivated by numerous, important quantum physics examples to introduce a framework for quantum symmetry breaking in terms of either locally compact quantum groupoid, or related algebroid, representations, such as those of weak Hopf C*-algebroids with convolution; the latter are usually realized in the context of rigged Hilbert spaces (Bohm and Gadella, 1989).

Furthermore, with regard to a unified and global framework for symmetry breaking, as well as higher order quantum symmetries, one needs to look towards the double groupoid structures of Brown and Spencer (1976), to enable one to introduce the concepts of quantum and graded Lie bi--algebroids which are expected to carry a distinctive C*--algebroid convolution structure. The extension to supersymmetry leads then naturally to superalgebra, superfield symmetries and their involvement in supergravity or quantum gravity (QG) theories for intense gravitational fields in fluctuating, quantized spacetimes. Their mathematical/quantum algebraic classification then involves superstructures with such supersymmetries that can only be appropriately studied in (quantum) supercategories.

Thus, a natural extension of quantum symmetries leads one to higher dimensional algebra (HDA) and may involve, for example, both `quantum' double groupoids defined as `locally compact' double groupoids equipped with Haar measures via convolution, and an extension of superalgebra to double (super) algebroids, (that are naturally much more general than the Lie double algebroids defined in Mackenzie, 2004).

One can now proceed to formally define several Quantum Algebraic Topology concepts that are needed to express the extended quantum symmetries in terms of proper quantum groupoid and algebroid representations. `Hidden', higher dimensional quantum symmetries will then also emerge either via generalized quantization procedures from higher dimensional algebra representations or be determined as global or local invariants obtainable-- at least in principle-- through non-Abelian algebraic topology (NAAT) methods.

Weak Hopf AlgebrasEdit

Let us begin by recalling the notion of a quantum group in relation to a Hopf algebra where the former is often realized as an automorphism group for a quantum space, that is, an object in a suitable category of generally noncommutative algebras. One of the most common guises of a quantum `group' is as the dual of a non-commutative, non-associative Hopf algebra. The Hopf algebras (cf. Chaician and Demichev 1996; Majid,1996), and their generalizations (Karaali, 2007), are some of the fundamental building blocks of quantum operator algebra (see the former's definition in the Appendix), even though they cannot be `integrated' to groups like the `integration' of Lie algebras to Lie groups; furthermore, the connection of Hopf algebras to quantum symmetries seems to be only indirect.

In order to define a weak Hopf algebra , one can relax certain axioms of a Hopf algebra as follows~:

 \item[(1)] The comultiplication is not necessarily unit--preserving.
 \item[(2)] The counit Failed to parse (unknown function "\vep"): {\displaystyle \vep}
 is not necessarily a homomorphism of algebras.

\item[(3)] The axioms for the antipode map Failed to parse (unknown function "\lra"): {\displaystyle S : A \lra A} with respect to the counit are as follows. For all ,

Failed to parse (unknown function "\ID"): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle {\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:

Failed to parse (unknown function "\xymatrix"): {\displaystyle \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}} }

Several mathematicians substitute the term \emph{quantum groupoid} for a weak Hopf algebra, although this algebra in itself is not a proper groupoid, but it may have a component group algebra as in the example of the quantum double discussed next; nevertheless, weak Hopf algebras generalize Hopf algebras --that with additional properties-- were previously introduced as quantum `groups' by mathematical physicists. (The latter are defined in the Appendix and, as already discussed, are not mathematical groups but algebras). As it will be shown in the next subsection, quasi--triangular quasi--Hopf algebras are directly related to quantum symmetries in conformal (quantum) field theories. Furthermore, weak C*--Hopf quantum algebras lead to weak C*--Hopf algebroids that are linked to quasi--group quantum symmetries, and also to certain Lie algebroids (and their associated Lie--Weinstein groupoids) used to define Hamiltonian (quantum) algebroids over the phase space of (quantum) --gravity.

Two Examples of Weak Hopf AlgebrasEdit

One can refer here to the example given by Bais et al. (2002). Let G be a non--Abelian group and a discrete subgroup. Let F(H) denote the space of functions on H and Failed to parse (unknown function "\bC"): {\displaystyle \bC H} the group algebra (which consists of the linear span of group elements with the group structure). The quantum double D(H) (Drinfel'd, 1987) is defined by the eqn :

Failed to parse (unknown function "\wti"): {\displaystyle D(H) = F(H)~ \wti{\otimes}~ \bC H~} , where, for , the `twisted tensor product' is specified by the next eqn:

</math>\wti{\otimes} \mapsto ~(f_1 \otimes h_1) (f_2 \otimes h_2)(x) = f_1(x) f_2(h_1 x h_1^{-1}) \otimes h_1 h_2 ~Failed to parse (syntax error): {\displaystyle . \ The physical interpretation given to this construction usually proceeds by considering <math>H} as the `electric gauge group', and F(H) as the `magnetic symmetry' generated by ~. In terms of the counit Failed to parse (unknown function "\vep"): {\displaystyle \vep} , the double D(H) has a trivial representation given by Failed to parse (unknown function "\vep"): {\displaystyle \vep(f \otimes h) = f(e)} ~. there are several very interesting features of this construction. For the purpose of braiding relations there is available an matrix, , leading to the following operator:

Failed to parse (unknown function "\a"): {\displaystyle \mathcal R \equiv \sigma \cdot (\Pi^A_{\a} \otimes \Pi^B_{\be})} to be defined in terms of the Clebsch--Gordan series Failed to parse (unknown function "\a"): {\displaystyle \Pi^A_{\a} \otimes \Pi^B_{\be} \cong N^{AB \gamma}_{\a \be C}~ \Pi^C_{\gamma}} , and where denotes a flip operator. The operator is sometimes called the monodromy or Aharanov--Bohm phase factor. In the case of a condensate in a state in the carrier space of some representation Failed to parse (unknown function "\a"): {\displaystyle \Pi^A_{\a}~} one considers the maximal Hopf subalgebra of a Hopf algebra for which is --invariant; specifically ~:

</math>\Pi^A_{\a} (P)~\vert v \rangle = \vep(P) \vert v \rangle~,~ \forall P \in T~.Failed to parse (unknown function "\U"): {\displaystyle For the second example, consider the example provided by Mack and Schomerus (1992) using a more general notion of the Drinfel'd construction--the notion of a \emph{quasi triangular quasi--Hopf algebra} (QTQHA) which was developed with the aim of studying a range of essential symmetries with special properties, such as the ''quantum group algebra'' <math>\U_q (\rm{sl}_2)} with ~. If , then it was shown that a QTQHA is canonically associated with Failed to parse (unknown function "\U"): {\displaystyle \U_q (\rm{sl}_2)} . Such QTQHAs are claimed as the true symmetries of minimal conformal field theories.

The Weak Hopf C*--Algebra in Relation to Quantum Symmetry BreakingEdit

In our setting,a Weak C*--Hopf algebra is 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' representation than any weak Hopf algebra. However, one can argue that locally compact groupoids equipped with a Haar measure (after quantization) come 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' . These transformations along with the coordinate space define certain Lie groupoids, and also their infinitesimal version - the Lie algebroids , when the former are Weinstein groupoids. If one then lifts the algebroid action from to the principal homogeneous space over the cotangent bundle Failed to parse (unknown function "\lra"): {\displaystyle T^*M \lra M} , one obtains a physically significant algebroid structure. The latter was called the Hamiltonian algebroid, , related to the Lie algebroid, . The Hamiltonian algebroid is an analog of the Lie Algebra of symplectic vector fields with respect to the canonical symplectic structure on or . In this recent example, the Hamiltonian algebroid, over , was defined over the phase space of --gravity, with the anchor map to Hamiltonians of canonical transformations (Levin and Olshanetsky, 2003,2008). Hamiltonian algebroids thus generalize Lie algebras of canonical transformations; canonical transformations of the Poisson sigma model phase space define a Hamiltonian algebroid with the Lie brackets related to such a Poisson structure on the target space. The Hamiltonian algebroid approach was utilized to analyze the symmetries of generalized deformations of complex structures on Riemann surfaces of genus with marked points. However, its implicit algebraic connections to von Neumann *--algebras and/or weak C*--algebroid representations have not yet been investigated. This example suggests that algebroid (quantum) symmetries are implicated in the foundation of relativistic quantum gravity theories and supergravity.

All SourcesEdit

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ReferencesEdit

  1. A. Abragam and B. Bleaney.: Electron Paramagnetic Resonance of Transition Ions. Clarendon Press: Oxford, (1970).
  2. E. M. Alfsen and F. W. Schultz: Geometry of State Spaces of Operator Algebras , Birkh\"auser, Boston--Basel--Berlin (2003).
  3. J.C. Baez and L. Langford. Higher-dimensional algebra. IV. 2-tangles. Adv. Math. 180 (no.2): 705-764 (2003).
  4. 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).
  5. I.C. Baianu: X--ray scattering by partially disordered membrane systems, Acta Cryst. A34 : 751-753 (1978).
  6. I.C. Baianu, K.A. Rubinson and J. Patterson. 1979. Ferromagnetic Resonance and Spin--Wave Excitations in Metallic Glasses., J. Phys. Chem. Solids , 40 : 940--951.
  7. I.C. Baianu, N. Boden,Y.K. Levine and D. Lightowlers. 1978. Dipolar Coupling between Groups of Three Spin--1/2 undergoing Hindered Reorientation in Solids., Solid-State Communications . 27: 474-478.
  8. I.C. Baianu, N. Boden, Y.K.Levine and S.M.Ross. 1978. Quasi--quadrupolar NMR Spin--Echoes in Solids Containing Dipolar Coupled Methyl Groups., J. Phys. (\textbf{C ) Solid State Physics}., 11 : L 37--41.
  9. I.C. Baianu, K.A. Rubinson and J. Patterson. 1979. The Observation of Structural Relaxation in a FeNiPB Glass by X--ray Scattering and Ferromagnetic Resonance., Physica Status Solidi (a), 53 K :133--135.
  10. I.C. Baianu, N. Boden and D. Lightowlers.1981. NMR Spin--Echo Responses of Dipolar--Coupled Spin--1/2 Triads in Solids., J. Magnetic Resonance , 43 :101-111.
  11. 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).
  12. I.C.Baianu, J.F. Glazebrook, and R. Brown.: On Physical Applications of Non-Abelian Algebraic Topology in Quantum Field Theories in preparation , (2008).
  13. I.C.Baianu, R. Brown J.F. Glazebrook, and G. Georgescu, Towards Quantum Non-Abelian Algebraic Topology. in preparation , (2008).
  14. 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).
  15. J.W. Barrett.: Geometrical measurements in three-dimensional quantum gravity., in: Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics (2001) . Intl. J. Modern Phys. A 18 , October, suppl., 97-113 (2003)
  16. A. D. Blaom : Lie Algebroids and Cartan's Method of Equivalence., arXiv:math/0509071v2 [math.DG], (12 Jan 2007).
  17. G. B\"ohm, F. Nill, and K. Szlach\'anyi: Weak Hopf algebras I, Integral theory and C*--structure, J. Algebra 221 (2): 385--438 (1999).
  18. A. Bohm and M. Gadella: Dirac kets, Gamow vectors and Gel'fand triples, Lect. Notes in Physics 348 Springer Verlag, 1989.
  19. Borceux,F. and G. Janelidze: Galois Theories , Cambridge Studies in Advanced Mathematics 72 , Cambridge (UK): Cambridge University Press, 2001.
  20. R. Brown R, K. Hardie, H. Kamps, T. Porter T.: The homotopy double groupoid of a Hausdorff space., Theory Appl. Categories , 10 :71-93 (2002).
  21. R. Brown: Topology and Groupoids , BookSurge LLC (2006).
  22. R. Brown R, P.J. Higgins, and R. Sivera.: 2008. Non--Abelian Algebraic Topology , PDF file download.
  23. R. Brown, J. F. Glazebrook and I. C. Baianu: A categorical and higher dimensional algebra framework for complex systems and spacetime structures, Axiomathes 17 :409--493. (2007).
  24. R. Brown and P. Higgins: tensor products and homotopies for  --groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987), 1--33.
  25. R. Brown and G. Janelidze.: Galois Theory and a new homotopy double groupoid of a map of spaces, Applied Categorical Structures. 12 : 63--80(2004).
  26. R. Brown and G. Janelidze: Van Kampen theorems for categories of covering morphisms in lextensive categories, J. Pure Appl. Algebra , 119 : 255-263, ISSN 0022--4049 (1997).
  27. 27.0 27.1 R. Brown and J.-L. Loday: Homotopical excision, and Hurewicz theorems, for n--cubes of spaces, Proc. London Math. Soc., 54:(3), 176- 192, (1987). Cite error: Invalid <ref> tag; name "BL87" defined multiple times with different content
  28. R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.
  29. R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. G\'eom.Diff. 17 (1976), 343-362.
  30. M. R. Buneci.: Groupoid Representations. , Ed. Mirton: Timisoara (2003).
  31. J. Butterfield and C. J. Isham : A topos perspective on the Kochen--Specker theorem I - IV, Int. J. Theor. Phys, 37 (1998) No 11., 2669--2733 38 (1999) No 3., 827--859, 39 (2000) No 6., 1413--1436, 41 (2002) No 4., 613-639.
  32. J. Butterfield and C. J. Isham : Some possible roles for topos theory in quantum theory and quantum gravity, Foundations of Physics., 4, 820---837 (2004).
  33. J.S. Carter and M. Saito. Knotted surfaces, braid moves, and beyond. Knots and quantum gravity (Riverside, CA, 1993), 191-229, Oxford Lecture Ser. Math. Appl., 1, Oxford Univ. Press, New York, 1994.
  34. M. Chaician and A. Demichev: Introduction to Quantum Groups , World Scientific (1996).
  35. Coleman and De Luccia: Gravitational effects on and of vacuum decay., Phys. Rev. D 21 : 3305 (1980).
  36. A. Connes: Noncommutative Geometry , Academic Press. 1994.
  37. 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).
  38. L. Crane; L.H. Kauffman; D.N. Yetter. State-sum invariants of 4-manifolds. J. Knot Theory Ramifications 6 , (no. 2): 177--234 1997).
  39. Day, B. J. and Street, R.: Monoidal bicategories and Hopf algebroids. Advances in Mathematics , 129 : 99--157 (1997)
  40. 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
  41. V. G. Drinfel'd: Quantum groups, In Proc. Int. Cong. of Mathematicians, Berkeley 1986 , (ed. A. Gleason), Berkeley, 798--820 (1987).
  42. G. J. Ellis: Higher dimensional crossed modules of algebras, J. of Pure Appl. Algebra 52 (1988), 277--282.
  43. P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups, Comm.Math.Phys., 196: 591\^a~@~S-640 (1998)
  44. P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19--52 (1999)
  45. 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.
  46. B. Fauser: A treatise on quantum Clifford Algebras. Konstanz, Habilitationsschrift., arXiv.math.QA/0202059 (2002).
  47. B. Fauser: Grade Free product Formulae from Grassman-Hopf Gebras. Ch. 18 in R. Ablamowicz, Ed., Clifford Algebras: Applications to Mathematics, Physics and Engineering , Birkh\"{a}user: Boston, Basel and Berlin, (2004).
  48. J. M. G. Fell.: The Dual Spaces of C*-Algebras, Transactions of the American Mathematical Society, 94 : 365-403 (1960).
  49. F.M. Fernandez and E. A. Castro.: (Lie) Algebraic Methods in Quantum Chemistry and Physics. , Boca Raton: CRC Press, Inc (1996).
  50. 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).]
  51. A.~Fr{\"o}hlich, Non-{A belian Homological Algebra. {I}. {D}erived functors and satellites.\/}, Proc. London Math. Soc. (3), 11: 239--252 (1961).
  52. Gel'fand, I. and Naimark, M., 1943, On the Imbedding of Normed Rings into the Ring of Operators in Hilbert Space, Recueil Math\'ematique [Matematicheskii Sbornik] Nouvelle S\'erie, 12 [54]: 197--213. [Reprinted in C*--algebras: 1943--1993, in the series Contemporary Mathematics, 167, Providence, R.I. : American Mathematical Society, 1994.]
  53. G. Georgescu: N--Valued Logics and \L ukasiewicz-Moisil Algebras., Axiomathes , 16 (1-2): 123-136 (March 2006).
  54. G. Georgescu, and D. Popescu.: On Algebraic Categories. Rev. Roum. Math. Pures et Appl .,24 (13):337--342 (1968).
  55. G. Georgescu, and C. Vraciu: On the Characterization of \L ukasiewicz Algebras., J. Algebra , 16(4): 486-495 (1970).
  56. R. Gilmore: "Lie Groups, Lie Algebras and Some of Their Applications." , Dover Publs., Inc.: Mineola and New York, 2005.
  57. J. Grabowski and G. Marmo: Generalized Lie bialgebroids and Jacobi structures, J. Geom Phys. 40 , no. 2, 176--199 (2001).
  58. M. Grandis and L. Mauri: Cubical sets and their site, \emph{Theor. Appl. Categories} 11 no. 38 (2003), 185--211.
  59. M. T. Grisaru and H. N. Pendleton: Some properties of scattering amplitudes in supersymmetric theories, Nuclear Phys. B 124 : 81--92(1977).
  60. Grothendieck, A.: 1957, Sur quelque point d-alg\'{e}bre homologique. , Tohoku Math. J. , 9: 119-121.
  61. Grothendieck, A. and J. Dieudon\'{e}.: 1960, El\'{e}ments de geometrie alg\'{e}brique., Publ. Inst. des Hautes Etudes de Science , 4 ; and Vol. III, Publ. Math. Inst. des Hautes Etudes , 11: 1-167 (1961).
  62. A. Grothendieck: Technique de descente et th\'eor\`emes d'existence en g\'eom\'etrie alg\'ebrique. II, S\'eminaire Bourbaki 12 (1959/1960), exp. 195, Secr\'etariat Math\'ematique, Paris, 1961.
  63. A. Grothendieck: Technique de construction en g\'eom\'etrie analytique. IV. Formalisme g\'en\'eral des foncteurs repr\^esentables, S\'eminaire H. Cartan 13 (1960/1961), exp. 11, Secr\'etariat Math\'ematique, Paris, 1962.
  64. Grothendieck, A.: 1971, Rev\^{e}tements \'Etales et Groupe Fondamental (SGA1), chapter VI: Cat\'egories fibr\'ees et descente, Lecture Notes in Math. 224 , Springer--Verlag: Berlin.
  65. P. Hahn: Haar measure for measure groupoids., Trans. Amer. Math. Soc . 242 : 1--33(1978).
  66. P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc . 242 :34--72(1978).
  67. R. Heynman and S. Lifschitz. 1958. Lie Groups and Lie Algebras., New York and London: Nelson Press.
  68. C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008) \\arXiv:0709.4364v2 [quant--ph]
  69. P.~J. {H}iggins, `Presentations of Groupoids, with Applications to Groups', Proc. {C amb. Phil. Soc.}60 : 7--20 (1964).
  70. P.~J. {H}iggins: Categories and Groupoids\/ , {v}an Nostrand: New York (1971).
  71. A. M. Hindeleh and R. Hosemann: Paracrystals representing the physical state of matter, Solid State Phys. 21 : 4155--4170 (1988).
  72. R. Hosemann and R. N. Bagchi: \emph{Direct Analysis of Diffraction by Matter}, North--Holland Publs.: Amsterdam--New York (1962).
  73. R. Hosemann, W. Vogel, D. Weick and F. J. Balta-Calleja: Novel aspects of the real paracrystal, Acta Cryst. A37 : 85--91 (1981).
  74. G. Janelidze.: `Precategories and Galois theory', Springer Lecture Notes in Math. 1488: 157-173(1991).
  75. G. Janelidze.: `Pure Galois theory in categories', J. Algebra , 132 : 270-286 (1990).
  76. V. Kac: Lie superalgebras, Advances in Math. 26 (1): 8--96 (1977).
  77. K.H. Kamps and T. Porter.: `A homotopy 2-groupoid from a fibration', Homotopy, homology and applications , 1 : 79-93(1999) .
  78. G. Karaali: On Hopf algebras and their generalizations. (2007) \\arXiv:math/07/03441v2
  79. J. Kustermans and S. Vaes: The operator algebra approach to quantum groups. Proc. Natl. Acad. USA , 97 , (2): 547-552, (2000).
  80. E. C. Lance: Hilbert C*--Modules. \emph{London Math. Soc. Lect. Notes} 210 , Cambridge Univ. Press. , (1995).
  81. N. P. Landsman: Mathematical topics between classical and quantum mechanics. Springer Verlag , New York, (1998).
  82. N. P. Landsman: Compact quantum groupoids, in `Quantum Theory and Symmetries' (Goslar, 18--22 July 1999) eds. H.-D. Doebner et al., World Scientific, (2000). \\arXiv:math-ph/9912006
  83. N. P. Landsman and B. Ramazan: Quantization of Poisson algebras associated to Lie algebroids. , In R. Kaminker et al, eds., Proc. Conf. on Groupoids in Physics, Analysis and Geometry , Contemporary Mathematics series, E--print: math--ph/001005.
  84. A. Levin and M. Olshanetsky.: Hamiltonian Algebroids and deformations of complex structures on Riemann curves., (2003, 2008). 
  85. J.-H. Lu: Hopf algebroids and quantum groupoids, Int. J. Math. 7 : 47--70 (1996).
  86. G. W. Mackey: The Scope and History of Commutative and Noncommutative Harmonic Analysis , Amer. Math. Soc., Providence, RI, (1992).
  87. G. Mack and V. Schomerus: Quasi Hopf quantum symmetry in quantum theory, Nucl. Phys. B 370 (1) (1992), 185--230.
  88. K. C. H. Mackenzie: \emph{General Theory of Lie Groupoids and Lie Algebroids}, London Math. Soc. Lecture Notes Series, 213 , Cambridge University Press, Cambridge, (2005).
  89. S. MacLane: "Categorical algebra." , Bull. Amer. Math. Soc. 71 , Number 1: 40-106(1965).
  90. S. MacLane and I. Moerdijk: \textit{Sheaves in Geometry and Logic - A first Introduction to Topos Theory}., Springer Verlag, New York (1992).
  91. S. Majid: Foundations of Quantum Group Theory , Cambridge Univ. Press (1995).
  92. G. Maltsiniotis.: Groupo des quantiques., C. R. Acad. Sci. Paris, 314 : 249 -- 252.(1992)
  93. J.P. May: A Concise Course in Algebraic Topology. Chicago and London: The Chicago University Press. (1999).
  94. B. Mitchell: The Theory of Categories , Academic Press, London, (1965).
  95. B. Mitchell: Rings with several objects., Adv. Math . 8 : 1 - 161 (1972).
  96. B. Mitchell: Separable Algebroids., Memoirs American Math. Soc. 333 :1x-1xx (1985).
  97. G. H. Mosa: \emph{Higher dimensional algebroids and Crossed complexes}, PhD thesis, University of Wales, Bangor, (1986).
  98. G. Moultaka, M. Rausch de Traubenberg and A. Tanas\u a: Cubic supersymmetry and abelain gauge invariance, \emph{Intl. J. Modern Phys. A} 20 no. 25 (2005), 5779-5806.
  99. J. Mr\v cun : On spectral representation of coalgebras and Hopf algebroids, (2002) preprint.\\  .
  100. P. Muhli, J. Renault and D. Williams: Equivalence and isomorphism for groupoid C*--algebras., J. Operator Theory , 17 : 3-22 (1987).
  101. D. A. Nikshych and L. Vainerman: J. Funct. Anal. 171 (2000) No. 2, 278--307
  102. H. Nishimura.: Logical quantization of topos theory., International Journal of Theoretical Physics , Vol. 35 ,(No. 12): 2555--2596 (1996).
  103. M. Neuchl, PhD Thesis, University of Munich (1997).
  104. V. Ostrik.: Module Categories over Representations of SL-q(2) in the Non--simple Case. (2006). \\ http://arxiv.org/PS--cache/math/pdf/0509/0509530v2.pdf
  105. A. L. T. Paterson: The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., Contemporary Mathematics 321 : 223-237 (2003)
  106. R. J. Plymen and P. L. Robinson : Spinors in Hilbert Space. Cambridge Tracts in Math. 114 , Cambridge Univ. Press (1994).
  107. N. Popescu.: The Theory of Abelian Categories with Applications to Rings and Modules. New York and London: Academic Press (1968).
  108. A. Ramsay: Topologies on measured groupoids., J. Functional Analysis , 47 : 314-343 (1982).
  109. A. Ramsay and M. E. Walter: Fourier-Stieltjes algebras of locally compact groupoids., J. Functional Analysis , 148 : 314-367 (1997).
  110. I. Raptis and R. R. Zapatrin : Quantisation of discretized spacetimes and the correspondence principle, \emph{Int. Jour. Theor. Phys.} 39 ,(1), (2000).
  111. T. Regge.: General relativity without coordinates. Nuovo Cimento (10 ) 19: 558-571 (1961).
  112. H.--K. Rehren: Weak C*--Hopf symmetry, \emph{Quantum Group Symposium at Group 21, Proceedings, Goslar (1996}, Heron Ptess, Sofia BG : 62--69(1997).
  113. J. Renault: A groupoid approach to C*--algebras. Lecture Notes in Maths. 793 , Berlin: Springer--Verlag,(1980).
  114. J. Renault: Representations de produits croises d'algebres de groupoides. , J. Operator Theory , 18 :67--97 (1987).
  115. J. Renault: The Fourier algebra of a measured groupoid and its multipliers., J. Functional Analysis , 145 : 455--490 (1997).
  116. M. A. Rieffel: Group C*--algebras as compact quantum metric spaces, Documenta Math. 7 : 605--651 (2002).
  117. M. A. Rieffel: Induced representations of C*--algebras, Adv. in Math. 13 : 176-254 (1974).
  118. J. E. Roberts : More lectures on algebraic quantum field theory (in: A. Connes, et al., Non--commutative Geometry ), Springer: Berlin (2004).
  119. J. Roberts.: Skein theory and Turaev-Viro invariants. Topology 34 ( no. 4): 771-S787 (1995).
  120. J. Roberts. Refined state-sum invariants of 3- and 4-manifolds. Geometric topology (Athens, GA, 1993), 217-234, AMS/IP Stud. Adv. Math ., 2.1 , Amer. Math.Soc., Providence, RI, (1997).
  121. Rovelli, C.: 1998, Loop Quantum Gravity , in N. Dadhich, et al. "Living Reviews in Relativity" \\ http:www.livingreviews.org/Articles/Volume1/1998--1--rovelli
  122. Schwartz, L.: Generalisation de la Notion de Fonction, de Derivation, de Transformation de Fourier et Applications Mathematiques et Physiques., Annales de l'Universite de Grenoble , 21 : 57--74 (1945).
  123. Schwartz, L.: Theorie des Distributions, \emph{Publications de l'Institut de Mathematique de l'Universit\'e de Strasbourg}, Vols 9--10, Paris: Hermann (1951--1952).
  124. 124.0 124.1 A. K. Seda: Haar measures for groupoids, \emph{Proc. Roy. Irish Acad. Sect. A} 76 No. 5, 25--36 (1976). Cite error: Invalid <ref> tag; name "Seda1" defined multiple times with different content
  125. A. K. Seda: Banach bundles of continuous functions and an integral representation theorem, Trans. Amer. Math. Soc. 270 No.1 : 327--332(1982).
  126. I.E. Segal.: Irreducible Representations of Operator Algebras. , Bulletin of the American Mathematical Society , 53 : 73--88 (1947a).
  127. I. E. Segal.: Postulates for General Quantum Mechanics, Annals of Mathematics , 4 : 930--948 (1947b).
  128. E.K. Sklyanin: Some Algebraic Structures Connected with the Yang-Baxter equation, Funct. Anal.Appl., 16 : 263--270 (1983).
  129. Some Algebraic Structures Connected with the Yang-Baxter equation. Representations of Quantum Algebras, Funct. Anal.Appl., 17: 273--284 (1984).
  130. K. Szlach\'anyi: The double algebraic view of finite quantum groupoids, J. Algebra 280 (1) , 249--294 (2004).
  131. M. E. Sweedler: Hopf algebras. W.A. Benjamin, INC., New York (1996).
  132. A. Tanas\u a: Extension of the Poincar\'e symmetry and its field theoretical interpretation, \emph{SIGMA Symmetry Integrability Geom. Methods Appl.} 2 (2006), Paper 056, 23 pp. (electronic).
  133. J.~{T}aylor, {\em Quotients of Groupoids by the Action of a Group\/}, Math. Proc. {C}amb. Phil. Soc., 103, (1988), 239--249.
  134. A.~P. Tonks, 1993, {\em Theory and applications of crossed complexes\/}, Ph.D. thesis, University of Wales, Bangor.
  135. V.G. Turaev and O.Ya. Viro. State sum invariants of 3--manifolds and quantum 6j--symbols. Topology 31 (no. 4): 865--902 (1992).
  136. E.~H.~van {Kampen}, {\em On the Connection Between the Fundamental Groups of some Related Spaces\/}, Amer. J. Math., 55, (1933), 261--267.
  137. P. Xu.: Quantum groupoids and deformation quantization. (1997) \\ arxiv.org/pdf/q--alg/9708020.pdf.
  138. D.N. Yetter.: TQFT' s from homotopy 2-types. J. Knot Theor . 2 : 113--123(1993).
  139. S. Weinberg.: The Quantum Theory of Fields . Cambridge, New York and Madrid: Cambridge University Press, Vols. 1 to 3, (1995--2000).
  140. A. Weinstein : Groupoids: unifying internal and external symmetry, Notices of the Amer. Math. Soc. 43 (7): 744--752 (1996).
  141. J. Wess and J. Bagger: Supersymmetry and Supergravity , Princeton University Press, (1983).
  142. 142.0 142.1 J. Westman: Harmonic analysis on groupoids, Pacific J. Math. 27 : 621-632. (1968). Cite error: Invalid <ref> tag; name "WJ1" defined multiple times with different content
  143. S. Wickramasekara and A. Bohm: Symmetry representations in the rigged Hilbert space formulation of quantum mechanics, J. Phys. A 35 (2002), no. 3, 807--829.
  144. Wightman, A. S., 1956, Quantum Field Theory in Terms of Vacuum Expectation Values., Physical Review, 101 : 860--866.
  145. Wightman, A.S.: 1976, Hilbert's Sixth Problem: Mathematical Treatment of the Axioms of Physics., In: Proceedings of Symposia in Pure Mathematics , 28 : 147--240.
  146. Wightman, A.S. and G\aa{}rding, L., 1964, {\em Fields as Operator--Valued Distributions in Relativistic Quantum Theory} , Arkiv fur Fysik, textbf{28}: 129--184.
  147. S. L. Woronowicz: Twisted SU(2) group : An example of a non--commutative differential calculus, RIMS, Kyoto University 23 (1987), 613--665.