12 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism the zeroth-iterative symplectic tensor is obtained as with the first-order symplectic potential given by This matrix is obviously singular, thus, it has a zero- mode that generates a constraint when contracted with the gradient of the potential U $^{(0)}$, given by, where U (1) = U $^{(0)}$. Therefore, the symplectic variables become ξ (1) α = ( φ, π$_{φ}$, A$_{0}$, A$_{1}$, π$_{1}$, η ) with the following one-form canonical momenta, U (1) = 1 2 ( π 2 1 + π 2 φ + φ ′ $^{2}$) − A$_{0}$ [ π ′ 1 + 1 2 q $^{2}$( a − 1) A$_{0}$ + q $^{2}$A$_{1}$ + qπ$_{φ}$ + qφ $^{′}$] − A$_{1}$ [ − qπ$_{φ}$ − 1 2 q $^{2}$( a + 1) A$_{1}$ − qφ $^{′}$] , (5.9) L (1) = π$_{φ}$ ˙ φ + π$_{1}$ ˙ A$_{1}$ + Ω$_{1}$ ˙ η − U $^{(1)}$, (5.8) f $^{(0)}$( x, y ) =    0 − 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 1 0   $^{δ}$$^{(}$$^{x}$ − y ) . (5.5) Ω$_{1}$ = ν (0) α ∂U (0) ∂ξ (0) α = π ′ 1 + q $^{2}$( a − 1) A$_{0}$ + q $^{2}$A$_{1}$ + qπ$_{φ}$ + qφ $^{′}$, (5.6) A (1) φ = π$_{φ}$, A (1) A$_{1}$ = π$_{1}$, A (1) A$_{0}$ = A (1) π$_{φ}$ = A (1) π$_{1}$ = 0 , (5.10) A (1) η = π ′ 1 + q $^{2}$( a − 1) A$_{0}$ + q $^{2}$A$_{1}$ + qπ$_{φ}$ + qφ $^{′}$. { Ω$_{1}$( x ) , Ω$_{1}$( y ) } = 0 . (5.7) f $^{(1)}$( x, y ) =    0 − 1 0 0 0 q∂$_{y}$ 1 0 0 0 0 q 0 0 0 0 0 q $^{2}$( a − 1) 0 0 0 0 − 1 q 2 0 0 0 1 0 ∂$_{y}$ − q∂$_{x}$ − q − q $^{2}$( a − 1) − q 2 − ∂$_{x}$ 0   $^{δ}$$^{(}$$^{x}$ − y ) , (5.11) { φ ( x ) , φ ( y ) } ∗ = 0 , { φ ( x ) , π$_{φ}$ ( y ) } ∗ = δ ( x − y ) , { φ ( x ) , A$_{0}$ ( y ) } ∗ = − 1 q ( a − 1) δ ( x − y ) , { φ ( x ) , A$_{1}$ ( y ) } ∗ = 0 , { φ ( x ) , π$_{1}$ ( y ) } ∗ = 0 , { π$_{φ}$ ( x ) , π$_{φ}$ ( y ) } ∗ = 0 , { π$_{φ}$ ( x ) , A$_{0}$ ( y ) } ∗ = 1 q ( a − 1) ∂$_{y}$δ ( x − y ) , { π$_{φ}$ ( x ) , A$_{1}$ ( y ) } ∗ = 0 , { π$_{φ}$ ( x ) , π$_{1}$ ( y ) } ∗ = 0 , { A$_{1}$ ( x ) , A$_{0}$ ( y ) } ∗ = − 1 q $^{2}$( a − 1) ∂$_{y}$δ ( x − y ) , (5.12) { A$_{1}$ ( x ) , A$_{1}$ ( y ) } ∗ = 0 , { A$_{1}$ ( x ) , π$_{1}$ ( y ) } ∗ = δ ( x − y ) , { π$_{1}$ ( x ) , A$_{0}$ ( y ) } ∗ = 1 ( a − 1) δ ( x − y ) , { π$_{1}$ ( x ) , π$_{1}$ ( y ) } ∗ = 0 . which is a nonsingular matrix. The inverse of f $^{(1)}$( x, y ) furnishes, after a straightforward calculation, the Dirac brackets among the physical fields, Substituting the constraint Ω$_{1}$ into the canonical sec- tor of the first-order Lagrangian through a Lagrange multiplier η , we have the first-iterative Lagrangian L $^{(1)}$, namely, The corresponding matrix f (1) is then that is identified as being the Gauss law, which satisfies the following Poisson algebra,