Gauged W Algebras

We perform an Hamiltonian reduction on a classical \cw(\cg, \ch) algebra, and prove that we get another \cw(\cg, \ch$'$) algebra, with $\ch\subset\ch'$. In the case $\cg=S\ell(n)$, the existence of a suitable gauge, called Generalized Horizontal Gauge, allows to relate in this way two \cw-algebras as soon as their corresponding \ch-algebras are related by inclusion.