Cohomological Destabilization of Casimir Invariants in q-Deformed Universal Enveloping Algebras: An Algebraic Mechanism for Spontaneous Translational Symmetry Breaking and Macroscopic Thrust Generation via Non-Involutive Automorphisms

Mathematical Applications of Science Fiction We investigate the stability of Casimir invariants within quantum groups under a specific class of non-involutive automorphisms. For generic complex deformation parameters that are not roots of unity, we demonstrate that the second Hochschild cohomology group of the quantum universal enveloping algebra admits a non-trivial obstruction class, preventing the Casimir element from remaining central under the twisted action of the translation generators. This cohomological destabilization appears as a non-vanishing commutator between the deformed momentum operator and the Hamiltonian, effectively breaking time-translational invariance in the local reference frame. We rigorously prove that this algebraic symmetry breaking produces a non-zero net flux vector in the representation space, which we interpret physically as macroscopic thrust generation without propellant—consistent with a modified conservation law arising from non-commutative geometry.