  It is shown that SU(N) gauge theory coupled to adjoint Higgs can be
explicitly re-written in terms of SU(N) gauge invariant dynamical variables
with local physical interactions. The resultant theory has a novel compact
abelian $U(1)^{(N - 1)}$ gauge invariance. The above abelian gauge invariance
is related to the adjoint Higgs field and not to the gauge group SU(N). In this
abelianized version the magnetic monopoles carrying the magnetic charges of
$(N-1)$ types have a natural origin and therefore appear explicitly in the
partition function as Dirac monopoles along with their strings. The gauge
invariant electric and magnetic charges with respect to $U(1)^{(N-1)}$ gauge
groups are shown to be vectors in root and co-root lattices of SU(N)
respectively. Therefore, the Dirac quantization condition corresponds to SU(N)
Cartan matrix elements being integers. We also study the effect of the $\theta$
term in the abelian version of the theory.
