2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/178090We exhibit the role of Hochschild cohomology in quantum field theory with particular emphasis on gauge theory and Dyson--Schwinger equations, the quantum equations of motion. These equations emerge from Hopf- and Lie algebra theory and free quantum field theory only. In the course of our analysis we exhibit an intimate relation between the Slavnov-Taylor identities for the couplings and the existence of Hopf sub-algebras defined on the sum of all graphs at a given loop order, surpassing the need to work on single diagrams.25p, uses feynmf, typos corrected, references added, to appear in Annals of PhysicsHigh Energy Physics - TheoryQuantum Algebratext