Hopf-Differential Algebraic Closure: A Unified Framework for Quantum Equation Solving

This paper establishes a comprehensive Hopf-differential algebraic framework that transcends the limitations of classical Galois theory. Our main innovation is the construction of the Hopf-differential algebraic closure KH of a Hopf algebra H, which provides explicit solutions tostructural equations in quantum algebras and representation theory. Extensive numerical validation demonstrates the effectiveness and accuracy of our approach. While traditional Galois theory prohibits radical solutions for general equations, we demonstrate that explicit analytic solutions exist in the appropriately extended Hopf-differential closure KH.