topic on the algebraic foundations of quantum algebraic topology

0.1 Topic entry on the algebraic foundations of Quantum Algebraic Topology.

(a). Quantum Algebraic Topology (QAT) is defined as the mathematical and physical study of general theories of quantum algebraic structures from the standpoint of Algebraic Topology, Category Theory and their Non-Abelian extensions in Higher Dimensional Algebra and Supercategories

(b). Several suggested new QAT topics are:

1.

Poisson algebras, Quantization methods and Hamiltonian algebroids
2.

K-S Theorem and its Quantum algebraic consequences in QAT
3.

Logic Lattice algebras or Many-Valued (MV) Logic algebras
4.

Quantum MV-Logic algebras and $\L{}-M_{n}$ -noncommutative lattice algebras
5.

Quantum Operator Algebras ( such as : involution, *-algebras, or $*$ -algebras, von Neumann algebras, , JB- and JL- algebras, $C^{*}$ - or C*- algebras,
6.

Quantum von Neumann algebra and subfactors
7.

Kac-Moody and K-algebras
8.

Hopf algebras, Quantum Groups and Quantum group algebras
9.

Quantum Groupoids and weak Hopf $C^{*}$ -algebras
10.

Groupoid C*-Convolution algebras and *-Convolution Algebroids
11.

Quantum Spacetimes and Quantum Fundamental Groupoids
12.

Quantum Double Algebras
13.

Quantum Gravity, supersymmetries, supergravity, superalgebras and graded ‘Lie’ algebras
14.

Quantum Categorical algebra and Higher Dimensional, $\L{}-M_{n}$ - Toposes
15.

Quantum R-categories, R-supercategories and Symmetry Breaking
16.

Extended Quantum Symmetries in Higher Dimensional Algebras (HDA), such as:
algebroids, double algebroids, categorical algebroids, double groupoids,
convolution algebroids, groupoid $C^{*}$ -convolution algebroids
17.

Universal algebras in R-Supercategories
18.

Supercategorical algebras (SA) as concrete interpretations of the Theory of Elementary Abstract Supercategories (ETAS).
19.

Quantum Non-Abelian Algebraic Topology (QNAAT)
20.

Noncommutative Geometry, Quantum Geometry, and Non-Abelian Quantum Algebraic Geometry
21.

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

References

1 Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkäuser, Boston–Basel–Berlin (2003).
2 Atyiah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France, 84: 307–317.
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.
8 Baez, J. & Dolan, J., 1998a, ”Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes”, Advances in Mathematics, 135, 145–206.
9 Baez, J. & Dolan, J., 2001, ”From Finite Sets 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 Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics, 32: 539-561.
12 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.
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 Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
20 Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.
21 Bell, J. L., 1981, ”Category Theory and the Foundations of Mathematics”, British Journal for the Philosophy of Science, 32, 349–358.
22 Bell, J. L., 1982, ”Categories, Toposes and Sets”, Synthese, 51, 3, 293–337.
23 Bell, J. L., 1986, ”From Absolute to Local Mathematics”, Synthese, 69, 3, 409–426.
24 Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press.
25 Birkoff, G. & Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.
26 Borceux, F.: 1994, Handbook of Categorical Algebra, vols: 1–3, in Encyclopedia of Mathematics and its Applications 50 to 52, Cambridge University Press.
27 Bourbaki, N. 1961 and 1964: Algèbre commutative., in Èléments de Mathématique., Chs. 1–6., Hermann: Paris.
28 BJk4) 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.
29 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
30 Brown, R., Glazebrook, J. F. and I.C. Baianu.: 2007b, A Conceptual, Categorical and Higher Dimensional Algebra Framework of Universal Ontology and the Theory of Levels for Highly Complex Structures and Dynamics., Axiomathes (17): 321–379.
31 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.
32 Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr., 71: 273-286.
33 Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, Cah. Top. Géom. Diff. 17, 343-362.
34 Brown R and Razak Salleh A (1999) Free crossed resolutions of groups and presentations of modules of identities among relations. LMS J. Comput. Math., 2: 25–61.
35 Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80: 1-34.
36 Buchsbaum, D. A.: 1969, A note on homology in categories., Ann. of Math. 69: 66-74.
37 Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179, 291-317.
38 Bunge, M., 1984, ”Toposes in Logic and Logic in Toposes”, Topoi, 3, no. 1, 13-22.
39 Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math, 179: 291-317.
40 Cartan, H. and Eilenberg, S. 1956. Homological Algebra, Princeton Univ. Press: Pinceton.
41 Cohen, P.M. 1965. Universal Algebra, Harper and Row: New York, London and Tokyo.
42 Connes A 1994. Noncommutative geometry. Academic Press: New York.
43 Croisot, R. and Lesieur, L. 1963. Algèbre noethérienne non-commutative., Gauthier-Villard: Paris.
44 Baianu, I. C. et al. 2008. Quantum Non-Abelian Algebraic Topology (QNAAT): PM Exposition lec $id=68$ .

[This is an entry-in-progress for a new topic entry on the Algebraic Foundations of Quantum Algebraic Topology.]

Title	topic on the algebraic foundations of quantum algebraic topology
Canonical name	TopicOnTheAlgebraicFoundationsOfQuantumAlgebraicTopology
Date of creation	2013-03-22 18:14:21
Last modified on	2013-03-22 18:14:21
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	34
Author	bci1 (20947)
Entry type	Topic
Classification	msc 08A99
Classification	msc 08A70
Classification	msc 18E05
Classification	msc 18-00
Classification	msc 55U40
Classification	msc 55-02
Classification	msc 81Q60
Classification	msc 08A05
Classification	msc 81R15
Synonym	Algebraic Foundations of Quantum Algebraic Topology topic
Related topic	QuantumAlgebraicTopology
Related topic	NonAbelianTheories
Related topic	GrassmanHopfAlgebrasAndTheirDualCoAlgebras