topic on algebraic foundations of quantum algebraic topology

This is a contributed topic on Quantum Algebraic Topology (QAT) introducing mathematical concepts of QAT based on algebraic topology (AT), category theory (CT) and their non-Abelian extensions in higher dimensional algebra (HDA) and supercategories.

0.1 Introduction

Quantum algebraic topology (QAT) is an area of physical mathematics and mathematical physics concerned with the foundation and study of general theories of quantum 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.

0.1.1 The following are examples of QAT topics:

Poisson algebras, quantization methods and Hamiltonian algebroids
K-S theorem and its quantum algebraic consequences in QAT
Logic lattice algebras and many-valued (MV) logic algebras
Quantum MV-logic algebras and Ł − M_n-noncommutative algebras
quantum operator algebras ( such as : involution, *-algebras, or ∗-algebras, von Neumann algebras, , JB- and JL- algebras, C^∗ - or C*- algebras,
Quantum von Neumann algebra and subfactors
Kac-Moody and K-algebras
quantum groups, quantum group algebras and Hopf algebras
quantum groupoids and weak Hopf C^∗-algebras
groupoid C*-convolution algebras and *-convolution algebroids
Quantum spacetimes and quantum fundamental groupoids
Quantum double algebras
quantum gravity, supersymmetries, supergravity, superalgebras and graded ‘Lie’ algebras
Quantum categorical algebra and higher dimensional, Ł − M_n- toposes
Quantum R-categories, R-supercategories and symmetry breaking
extended quantum symmetries in higher dimensional algebras (HDA), such as: algebroids, double algebroids, categorical algebroids, double groupoids ,convolution algebroids, and groupoid C^∗ -convolution algebroids
Universal algebras in R-supercategories
Supercategorical algebras (SA) as concrete interpretations of the theory of elementary abstract supercategories (ETAS).
Non-Abelian quantum algebraic topology (NAQAT)
noncommutative geometry, quantum geometry, and non-Abelian quantum algebraic geometry
Kochen-Specker theorem (K-S theorem)
Other – Miscellaneous

References

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[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 Łukasiewicz 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.

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[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: Algèbre commutative., in Éléments de Mathématique., 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. Géom. 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] Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. 80: 1-34.

[25] Buchsbaum, D. A.: 1969, A note on homology in categories., Ann. of Math. 69: 66-74.

[26] Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, Adv. in Math. 179, 291-317.

[27] Bunge, M., 1984, Toposes in Logic and Logic in Toposes, Topoi, 3, no. 1, 13-22.

[28] Bunge M, Lack S (2003) Van Kampen theorems for toposes. Adv Math, 179: 291-317.

[29] Cartan, H. and Eilenberg, S. 1956. Homological Algebra, Princeton Univ. Press: Pinceton.

[30] Cohen, P.M. 1965. Universal Algebra, Harper and Row: New York, London and Tokyo.

[31] Connes A 1994. Noncommutative geometry. Academic Press: New York.

[32] Croisot, R. and Lesieur, L. 1963. Algèbre noethérienne non-commutative., Gauthier-Villard: Paris.

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