topic on the Algebraic Foundations of Quantum Algebraic Topology

topic on the Algebraic Foundations of Quantum Algebraic Topology

topic on the Algebraic Foundations of Quantum Algebraic Topology



This is a contributed topic on the algebraicMathworldPlanetmathPlanetmath foundations of Quantum Algebraic Topology (QAT)

(A.) Quantum Algebraic Topology (QAT) is defined as the mathematical and physical study of general theories of quantum algebraic structuresPlanetmathPlanetmath from the standpoint of Algebraic Topology, Category TheoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath and their Non-AbelianMathworldPlanetmathPlanetmath extensionsPlanetmathPlanetmathPlanetmath in Higher Dimensional AlgebraPlanetmathPlanetmath and SupercategoriesPlanetmathPlanetmath in relationMathworldPlanetmathPlanetmathPlanetmath to, or petinent to, Quantum theoriesPlanetmathPlanetmath, Quantum Field Theories, General Relativity and its Quantum extensions, Quantum Gravity.

(B). Several suggested new QAT topics are:

  1. (1)

    Poisson algebras, Quantization methods and Hamiltonian algebroids

  2. (2)

    K-S Theorem and its Quantum algebraic consequences in QAT

  3. (3)

    Logic Lattice algebras or Many-Valued (MV) Logic algebras

  4. (4)

    Quantum MV-Logic algebrasMathworldPlanetmathPlanetmath and Ł-Mn-noncommutative algebras

  5. (5)

    Quantum Operator AlgebrasPlanetmathPlanetmathPlanetmath ( such as : involutionPlanetmathPlanetmath, *-algebras, or *-algebras, von Neumann algebrasMathworldPlanetmathPlanetmathPlanetmath, JB- and JL- algebras, C* - or C*- algebras, etc.

  6. (6)

    Quantum von Neumann algebra and subfactors

  7. (7)

    Kac-Moody and K-algebras

  8. (8)

    Hopf algebrasMathworldPlanetmathPlanetmath, Quantum GroupsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and Quantum group algebras

  9. (9)

    Quantum GroupoidsPlanetmathPlanetmath and weak Hopf C*-algebras

  10. (10)

    GroupoidPlanetmathPlanetmathPlanetmathPlanetmath C*-Convolution algebras and *-Convolution Algebroids

  11. (11)

    Quantum Spacetimes and Quantum Fundamental Groupoids

  12. (12)

    Quantum Double Algebras

  13. (13)

    Quantum Gravity, supersymmetries, supergravity, superalgebras and graded ‘Lie’ algebras

  14. (14)

    Quantum Categorical algebra and Higher Dimensional, Ł-Mn- Toposes

  15. (15)

    Quantum R-categories, R-supercategories and Symmetry Breaking

  16. (16)

    Extended Quantum Symmetries in Higher Dimensional Algebras (HDA), such as:
    algebroids, double algebroids, categorical algebroids, double groupoidsPlanetmathPlanetmathPlanetmath,
    convolution algebroids, groupoid C* -convolution algebroids

  17. (17)

    Universal algebras in R-Supercategories

  18. (18)

    Supercategorical algebras (SA) as concrete interpretationsMathworldPlanetmathPlanetmath of the Theory of Elementary Abstract Supercategories (ETAS).

  19. (19)

    Quantum Non-Abelian Algebraic Topology (QNAAT)

  20. (20)
  21. (21)

    Other – Miscellaneous [please add here your additionsPlanetmathPlanetmath, changes, editing, remarks, proofs, conjectures, and so on…]

References

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  • 3 Auslander, M. 1965. Coherent Functors. Proc. Conf. Cat. Algebra, La Jolla, 189–231.
  • 4 Awodey, S. & Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168–1182.
  • 5 Awodey, S., 1996, ”Structure in Mathematics and Logic: A Categorical Perspective”, Philosophia Mathematica, 3, 209–237.
  • 6 Awodey, S., 2004, ”An Answer to Hellman’s Question: Does Category Theory Provide a Framework for Mathematical Structuralism”, Philosophia Mathematica, 12, 54–64.
  • 7 Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.
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  • 9 Baez, J. & Dolan, J., 2001, ”From Finite SetsMathworldPlanetmath to Feynman Diagrams”, Mathematics Unlimited – 2001 and Beyond, Berlin: Springer, 29–50.
  • 10 Baez, J., 1997, ”An Introduction to n-Categories”, Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1–33.
  • 11 Baianu, I.C.: 1970, Organismic SupercategoriesPlanetmathPlanetmath: II. On Multistable Systems. Bulletin of Mathematical Biophysics, 32: 539-561.
  • 12 Baianu, I.C.: 1971b, CategoriesMathworldPlanetmath, FunctorsMathworldPlanetmath and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science, September 1–4, 1971, Bucharest.
  • 13 Baianu, I.C. and D. Scripcariu: 1973, On Adjoint Dynamical Systems. Bulletin of Mathematical Biophysics, 35(4), 475–486.
  • 14 Baianu, I.C.: 1973, Some Algebraic Properties of (M,R) – Systems. Bulletin of Mathematical Biophysics 35, 213-217.
  • 15 Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology, 39: 249-258.
  • 16 Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)–Systems and Their Higher Dimensional Algebra, Abstract and Preprint of Report: http://www.ag.uiuc.edu/fs401/QAuto.pdf and http://www.medicalupapers.com/quantum+automata+math+categories+baianu/
  • 17 Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007a, Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness, Axiomathes, 17: 35-168.
  • 18 Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
  • 19 Baianu, I. C. et al. 2008. Quantum Non-Abelian Algebraic Topology (QNAAT): PM Exposition lec i⁢d=68.
  • 20 Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
  • 21 Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
  • 22 Bell, J. L., 1981, ”Category Theory and the Foundations of Mathematics”, British Journal for the Philosophy of Science, 32, 349–358.
  • 23 Bell, J. L., 1982, ”Categories, Toposes and Sets”, Synthese, 51, 3, 293–337.
  • 24 Bell, J. L., 1986, ”From Absolute to Local Mathematics”, Synthese, 69, 3, 409–426.
  • 25 Bell, J. L., 1988, Toposes and Local Set TheoriesMathworldPlanetmath: An Introduction, Oxford: Oxford University Press.
  • 26 Birkoff, G. & Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.
  • 27 Borceux, F.: 1994, Handbook of Categorical Algebra, vols: 1–3, in Encyclopedia of Mathematics and its Applications 50 to 52, Cambridge University Press.
  • 28 Bourbaki, N. 1961 and 1964: Algèbre commutativePlanetmathPlanetmathPlanetmath., in Èléments de Mathématique., Chs. 1–6., Hermann: Paris.
  • 29 Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoidPlanetmathPlanetmath of a map of spaces, Applied Categorical Structures 12: 63-80.
  • 30 Brown, R., Higgins, P. J. and R. Sivera,: 2007a, Non-Abelian Algebraic Topology, in preparation.
    http://www.bangor.ac.uk/ mas010/nonab-a-t.html ;
    http://www.bangor.ac.uk/ mas010/nonab-t/partI010604.pdf
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  • 33 Brown, R., and Hardy, J.P.L.:1976, Topological groupoidsPlanetmathPlanetmathPlanetmathPlanetmath I: universal constructions, Math. Nachr., 71: 273-286.
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Title topic on the Algebraic Foundations of Quantum Algebraic Topology
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Date of creation 2013-03-11 19:51:44
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