Quantum Symmetries, Operator Algebra and Quantum Groupoid Representations: Paracrystalline Systems, Topological Order, Supersymmetry and Global Symmetry Breaking

Novel approaches to extended quantum symmetry, paracrystals, quasicrystals, noncrystalline solids, topological order, supersymmetry and spontaneous, global symmetry breaking are outlined in terms of quantum groupoid, quantum double groupoids and dual, quantum algebroid structures. Physical applications of such quantum groupoid and quantum algebroid representations to quasicrystalline structures and paracrystals, quantum gravity, as well as the applications of the Goldstone and Noether's theorems to: phase transitions in superconductors/superfluids, ferromagnets, antiferromagnets, mictomagnets, quasi-particle (nucleon) ultra-hot plasmas, nuclear fusion, and the integrability of quantum systems are also considered. Both conceptual developments and novel approaches to Quantum theories are here proposed starting from existing Quantum Group Algebra (QGA), Algebraic Quantum Field Theories (AQFT), standard and effective Quantum Field Theories (QFT), as well as the refined `machinery' of Non--Abelian Algebraic Topology (NAAT), Category Theory (CT) and Higher-Dimensional Algebra (HDA).The logical links between Quantum Operator Algebras and their corresponding,'dual' structure of the Quantum State Spaces are also investigated. Among the key concepts presented are: Quantum Group Algebras (QGAs)/Groupoids, Hopf and C*algebras, Lie `algebras', Quantization and Asymptotic Morphisms, Locally Topological Groupoids, Crossed Modules of Groups or Lie Double Groupoids, Lie Algebroids, Crossed Complexes over Groupoids, Holonomy and Gauge Transformation Groupoids, Quantum Principal Bundles and Sheaves.