2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/50710Renormalization is cast in the form of a Lie algebra of infinite triangular matrices. By exponentiation, these matrices generate counterterms for Feynman diagrams with subdivergences. As representations of an insertion operator, the matrices are related to the Connes-Kreimer Lie algebra. In fact, the right-symmetric nonassociative algebra of the Connes-Kreimer insertion product is equivalent to an "Ihara bracket" in the matrix Lie algebra. We check our results in a three-loop example in scalar field theory. Apart from possible applications in high-precision phenomenology, we give a few ideas about possible applications in noncommutative geometry and functional integration.32 pages, uses feynmf package. v2: added appendix, corrected typosHigh Energy Physics - Theorytext