Noncommutative Physics on Lie Algebras, (ℤ2) n Lattices and Clifford Algebras

We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type cocycle twist, such as the noncommutative torus, θ-spaces and Clifford algebras. The latter are noncommutative deformations of the finite lattice (ℤ₂)n and we compute their noncommutative de Rham cohomology and moduli of solutions of Maxwell’s equations. We exactly quantize noncommutative U(1)-Yang-Mills theory on ℤ₂ × ℤ₂ in a path integral approach.