General solution of the non-Abelian Gauss law and non-Abelian analogues of the Hodge decomposition

A general solution of the non-Abelian Gauss law in terms of covariant curls and gradients is presented. Also two non-Abelian analogues of the Hodge decomposition in three dimensions are addressed: (i) A decomposition of an isotriplet vector field $V_i^a\mathrm{}\left(x\right)\mathrm{}$ as the sum of a covariant curl and gradient with respect to an arbitrary background Yang-Mills potential is obtained; (ii) a decomposition of the form $V_i^a{=B}_i^a{\mathrm{}\left(C\right)+D}_i\mathrm{}\left(C\right)\mathrm{}{\phi }^a$ which involves a non-Abelian magnetic field of a new Yang-Mills potential C is also presented. These results are relevant for duality transformation for non-Abelian gauge fields.