  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. These include: controlled
nuclear fusion and other nuclear reaction 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 and
supergravity. 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. Fourier transforms, generalized Fourier-Stieltjes transforms, and
duality relations link, respectively, the quantum groups and quantum groupoids
with their dual algebraic structures; quantum double constructions are also
discussed in this context in relation to quasi-triangular, quasi-Hopf algebras,
bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the
one hand, this quantum algebraic approach is known to provide solutions to the
quantum Yang-Baxter equation. On the other hand, our novel approach to extended
quantum symmetries and their associated representations is shown to be relevant
to locally covariant general relativity theories that are consistent with
either nonlocal quantum field theories or local bosonic (spin) models with the
extended quantum symmetry of entangled, 'string-net condensed' (ground) states.
