PlanetPhysics/Topic on Algebraic Foundations of Quantum Algebraic Topology

From testwiki
Revision as of 16:13, 12 September 2020 by imported>MaintenanceBot (Cleanup)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

This is a contributed topic on [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]] (QAT) introducing mathematical [[../PreciseIdea/|concepts]] of QAT based on [[../ModuleAlgebraic/|algebraic topology]] (AT), [[../TrivialGroupoid/|category theory]] (CT) and their [[../NonAbelianQuantumAlgebraicTopology3/|non-Abelian]] extensions in [[../InfinityGroupoid/|higher dimensional algebra]] ([[../2Groupoid2/|HDA]]) and [[../SuperCategory6/|supercategories]].

Introduction

Quantum algebraic topology (QAT) is an area of [[../IHES/|physical mathematics]] and [[../PhysicalMathematics2/|mathematical physics]] concerned with the foundation and study of [[../GeneralTheory/|general theories]] of quantum [[../TrivialGroupoid/|algebraic structures]] from the standpoint of algebraic topology, category theory, as well as non-Abelian extensions of AT and CT in higher dimensional algebra and supercategories.

The following are examples of QAT topics:

  1. [[../PoissonRing/|Poisson algebras]], [[../QuantizationMethods/|quantization methods]] and [[../Algebroids/|Hamiltonian algebroids]]
  1. K-S theorem and its quantum [[../CoIntersections/|algebraic]] consequences in QAT
  1. Logic lattice algebras and many-valued (MV) logic algebras
  1. Quantum MV-logic algebras and Failed to parse (unknown function "\L"): {\displaystyle \L{}-M_n} -noncommutative algebras
  1. [[../Groupoid/|quantum operator algebras]] ( such as : involution, *-algebras, or *-algebras, [[../VonNeumannAlgebra2/|von Neumann algebras]],

, JB- and JL- algebras, C* - or C*- algebras,

  1. Quantum von Neumann algebra and subfactors
  1. Kac-Moody and K-algebras
  1. [[../ComultiplicationInAQuantumGroup/|quantum groups]], quantum group algebras and [[../Groupoid/|Hopf algebras]]
  1. [[../WeakHopfAlgebra/|quantum groupoids]] and weak Hopf C*-algebras
  1. [[../GroupoidCConvolutionAlgebra/|groupoid C*-convolution algebras]] and *-convolution [[../Algebroids/|algebroids]]
  2. [[../QuantumSpaceTimes/|Quantum spacetimes]] and [[../QuantumFundamentalGroupoid4/|quantum fundamental groupoids]]
  1. Quantum double algebras
  1. [[../LQG2/|quantum gravity]], [[../Supersymmetry/|supersymmetries]], [[../AntiCommutationRelations/|supergravity]], [[../MathematicalFoundationsOfQuantumTheories/|superalgebras]] and graded `[[../BilinearMap/|Lie' algebras]]
  2. Quantum [[../CategoryOfLogicAlgebras/|categorical algebra]] and higher dimensional, Failed to parse (unknown function "\L"): {\displaystyle \L{}-M_n} - toposes
  1. Quantum [[../RCategory/|R-categories]], [[../RDiagram/|R-supercategories]] and symmetry breaking
  1. [[../ExtendedQuantumSymmetries/|extended quantum symmetries]] in higher dimensional algebras (HDA), such as:

algebroids, [[../GeneralizedSuperalgebras/|double algebroids]], categorical algebroids, [[../WeakHomotopy/|double groupoids]][[../AssociatedGroupoidAlgebraRepresentations/|,convolution]] algebroids, and [[../LocallyCompactGroupoid/|groupoid]] C* -convolution algebroids

  1. Universal algebras in R-supercategories
  1. Supercategorical algebras (SA) as concrete interpretations of the theory of elementary abstract supercategories ([[../ETACAxioms/|ETAS]]).
  1. Non-Abelian quantum algebraic topology (NAQAT)
  2. [[../NoncommutativeGeometry4/|noncommutative geometry]], quantum geometry, and non-Abelian quantum algebraic geometry
  3. Kochen-Specker theorem (K-S theorem)
  4. Other -- Miscellaneous

\begin{thebibliography} {9} </ref>[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][24][25][26][27][28][29][30][31]</references>

Template:CourseCat

  1. ↑ Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras , Birk\"auser, Boston--Basel--Berlin (2003).
  2. ↑ Atiyah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France , 84 : 307--317.
  3. ↑ Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.
  4. ↑ Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, Advances in Mathematics, 135, 145--206.
  5. ↑ Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, Mathematics Unlimited -- 2001 and Beyond, Berlin: Springer, 29--50.
  6. ↑ Baez, J., 1997, An Introduction to n-Categories, Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1--33.
  7. ↑ Baianu, I.C.: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science , September 1--4, 1971, Bucharest.
  8. ↑ Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R) --Systems and Their Higher Dimensional Algebra, Abstract and Preprint of Report
  9. ↑ Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
  10. ↑ Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
  11. ↑ Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
  12. ↑ Bell, J. L., 1981, Category Theory and the Foundations of Mathematics, British Journal for the Philosophy of Science, 32, 349--358.
  13. ↑ Bell, J. L., 1982, Categories, Toposes and Sets, Synthese, 51, 3, 293--337.
  14. ↑ Bell, J. L., 1986, From Absolute to Local Mathematics, Synthese, 69, 3, 409--426.
  15. ↑ Birkoff, G. \& Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.
  16. ↑ Borceux, F.: 1994, Handbook of Categorical Algebra , vols: 1--3, in Encyclopedia of Mathematics and its Applications 50 to 52 , Cambridge University Press.
  17. ↑ Bourbaki, N. 1961 and 1964: Alg\`{e bre commutative.}, in \'{E}l\'{e}ments de Math\'{e}matique., Chs. 1--6., Hermann: Paris.
  18. ↑ Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoid of a map of spaces, Applied Categorical Structures 12 : 63-80.
  19. ↑ Brown, R., Higgins, P. J. and R. Sivera,: 2008, Non-Abelian Algebraic Topology , (vol.2 in preparation).
  20. ↑ Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., Theory and Applications of Categories 10 , 71-93.
  21. ↑ Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr. , 71: 273-286.
  22. ↑ Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, Cah. Top. G\'{e om. Diff.} 17 , 343-362.
  23. ↑ Brown R, Razak Salleh A (1999) Free crossed resolutions of groups and presentations of modules of identities among relations. LMS J. Comput. Math. , 2 : 25--61.
  24. ↑ 24.0 24.1 Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80 : 1-34. Cite error: Invalid <ref> tag; name "BDA55" defined multiple times with different content
  25. ↑ Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179 , 291-317.
  26. ↑ Bunge, M., 1984, Toposes in Logic and Logic in Toposes, Topoi , 3, no. 1, 13-22.
  27. ↑ Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math , \textbf {179}: 291-317.
  28. ↑ Cartan, H. and Eilenberg, S. 1956. Homological Algebra , Princeton Univ. Press: Pinceton.
  29. ↑ Cohen, P.M. 1965. Universal Algebra , Harper and Row: New York, London and Tokyo.
  30. ↑ Connes A 1994. Noncommutative geometry . Academic Press: New York.
  31. ↑ Croisot, R. and Lesieur, L. 1963. Alg\`ebre noeth\'erienne non-commutative. , Gauthier-Villard: Paris.
Retrieved from "https://en.wikiversity.beta.math.wmflabs.org/w/index.php?title=PlanetPhysics/Topic_on_Algebraic_Foundations_of_Quantum_Algebraic_Topology&oldid=2224"