  We 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.
