  We obtain the exact Dirac algebra obeyed by the conserved non-local charges
in bosonic non-linear sigma models. Part of the computation is specialized for
a symmetry group $O(N)$. As it turns out the algebra corresponds to a cubic
deformation of the Kac-Moody algebra. The non-linear terms are computed in
closed form. In each Dirac bracket we only find highest order terms (as
explained in the paper), defining a saturated algebra. We generalize the
results for the presence of a Wess-Zumino term. The algebra is very similar to
the previous one, containing now a calculable correction of order one unit
lower.
