  Following the Schwinger boson representation for the su(M+1)- and the
su(N,1)-algebra presented by two of the present authors (J. da P. and M. Y.)
and Kuriyama, a possible counterpart of the Lipkin model in the su(M+1)-algebra
formulated in the fermion space is presented. The free vacuum, which plays a
fundamental role in the conventional treatment of the Lipkin model, is
generalized in a quite natural way, and further, the excited state generating
operators such as the particle-hole pairs are also given in a natural scheme.
As concrete examples, the cases of the su(2)-, su(3)- and the su(4)-algebra are
discussed. Especially, the case of the su(4)-algebra is investigated in detail
in relation to the nucleon pairing correlations and the high temperature
superconductivity.
