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Updated version: November 20, 2008, 61 pages, 9 Sections, 178 references

Abstract:

A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications (such as, controlled nuclear fusion and other nuclear reactions studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions, supergravity, and so on). This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non--Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. The relevance of our approach to extended quantum symmetries and their associated representations in locally covariant General Relativity theories that are consistent with nonlocal quantum field theories are also presented. *** Keywords: Extended Quantum Symmetries, Groupoids and Algebroids; Quantum Algebraic Topology (QAT); Algebraic Topology of Quantum Systems; Symmetry Breaking, Paracrystals, Superfluids, Spin Networks and Spin Glasses; Convolution Algebras and Quantum Algebroids; Nuclear Frech\'et spaces and GNS Representations of Quantum State Spaces (QSS); Groupoid and Functor Representations in relation to extended quantum symmetries in QAT; Quantization procedures; Quantum Algebras: Von Neumann algebra Factors, Paragroups and Kac algebras; Quantum Groups and Ring structures; Lie algebras, Lie Algebroids, Grassmann-Hopf, Weak C*-Hopf and Graded Lie algebras, Weak C*-Hopf algebroids; Compact Quantum Groupoids; Quantum Groupoid C*-algebras; Relativistic Quantum Gravity (RQG), Supergravity and Supersymmetry theories; Fluctuating Quantum spacetimes; Intense Gravitational Fields; Hamiltonian Algebroids in Quantum Gravity; Poisson-Lie Manifolds and Quantum Gravity Theories; Quantum Fundamental Groupoids; Tensor Products of Algebroids and Categories; Quantum Double Groupoids and Algebroids; Higher Dimensional Quantum Symmetries; Applications of Generalized van Kampen Theorem (GvKT) to Quantum Spacetime invariants. Subjclass--Mathematics Classification: Primary: msc:18B40,81R50,22A22,46L05; Secondary:55U40,81R10,05C38,22D25,05A15,15A18,81T05,43A25,43A35,46L87,15A15

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Open access Copyright@2008 by I.C. Baianu, J.F. Galzebrook and R. Brown http://www.bangor.ac.uk/~mas010/pdffiles/BBG-158-NAC0STQG.pdf http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/07/cathom07.html http://www.bangor.ac.uk/~mas0

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