Preprint/ preview:
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.
