  In the conventional formulation of N=1 supersymmetry, a vector multiplet is
supposed to be in the adjoint representation of a given gauge group. We present
a new formulation with a vector multiplet in the non-adjoint representation of
SO(N) gauge group. Our basic algebra is [ T^I, T^J ] = f^{I J K} T^K, [ T^I,
U^i ] = - (T^I)^{i j} U^j, [ U^i, U^j ] = - (T^I)^{i j} T^I, where T^I are the
generators of SO(N), while U^i are the new 'generators' in certain non-adjoint
real representation R of SO(N). We use here the word `generator' in the broader
sense of the word. Such a representation can be any real representation of
SO(N) with the positive definite metric, satisfying (T^I)^{i j} = - (T^I)^{j i}
and (T^I)^{[ i j |} (T^I)^{| k ] l} \equiv 0. The first non-trivial examples
are the spinorial 8_S and conjugate spinorial 8_C representations of SO(8)
consistent with supersymmetry. We further couple the system to chiral
multiplets, and show that a Higgs mechanism can give positive definite (mass)^2
to the new gauge fields for U^i. We show an analogous system working with N=1
supersymmetry in 10D, and thereby N=4 system in 4D interacting with extra
multiplets in the representation R. We also perform superspace reformulation as
an independent confirmation.
