2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152615We present an analysis of the cocycle appearing in the vertex operator representation of simply-laced, affine, Kac-Moody algebras. We prove that it can be described in the context of $R$-commutative geometry, where $R$ is a Yang-Baxter operator, as a strong $R$-commutative algebra. We comment on the Hochschild, cyclic and dihedral homology theories that appear in non-commutative geometry and their potential relation to string theory.26 pagesHigh Energy Physics - TheoryQuantum Algebratext