More about hidden-symmetry algebra for the self-dual Yang-Mills system

The infinite-parameter "hidden symmetry" algebra for the complex self-dual $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ Yang-Mills fields is extended from $\mathrm{SL}(N, C)\otimes C\left[\lambda \right]$ to $\mathrm{SL}(N, C)\otimes C[\lambda , {\lambda }^{-1\mathrm{}}]\otimes \mathrm{SL}(N, C)$. Furthermore, we also find an infinite set of infinitesimal "hidden symmetry" transformations, indexed by all integers, which can maintain the reality of the self-dual Yang-Mills potentials. Their infinite-parameter Lie algebra is shown to be related to an infinitedimensional symmetric space with isotropy group $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}\otimes R\left[\lambda \right]$.