Abstract
The infinite-parameter "hidden symmetry" algebra for the complex self-dual $\mathrm{SU}(N)$ Yang-Mills fields is extended from $\mathrm{SL}(N, C)\ensuremath{\bigotimes}C[\ensuremath{\lambda}]$ to $\mathrm{SL}(N, C)\ensuremath{\bigotimes}C[\ensuremath{\lambda}, {\ensuremath{\lambda}}^{\ensuremath{-}1}]\ensuremath{\bigotimes}\mathrm{SL}(N, C)$. 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 $\mathrm{SU}(N)\ensuremath{\bigotimes}R[\ensuremath{\lambda}]$.
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