  We study the canonical structure of the $SU(N)$ non-linear Sigma-model in a
polynomial, first-order representation. The fundamental variables in this
description are a non-Abelian vector field L_mu and a non-Abelian antisymmetric
tensor field theta_{mu nu}, which constrains L_{mu} to be a `pure gauge' (F_{mu
nu}(L) = 0) field. The second-class constraints that appear as a consequence of
the first-order nature of the Lagrangian are solved, and the reduced
phase-space variables explicitly found. We also treat the first-class
constraints due to the gauge-invariance under transformations of the
antisymmetric tensor field, constructing the corresponding most general
gauge-invariant functionals, which are used to describe the dynamics of the
physical degrees of freedom. We present these results in 1+1, 2+1 and 3+1
dimensions, mentioning some properties of the (d+1)-dimensional case. We show
that there is a kind of duality between this description of the non-linear
$\sigma$-model and the massless Yang-Mills theory. This duality is further
extended to more general first-class systems.
