  We use the idea of partial Gauss decomposition to study structures related to
$U_q(\widehat{{{\frak{gl}}}(n-1)})$ inside $U_q(\widehat{{{\frak{gl}}}(n)}) $.
This gives a description of $U_q(\widehat{{{\frak{gl}}}(n)})$ as an extension
of $U_q(\widehat{{{\frak{gl}}}(n-1)})$ with Zamolodchikov algebras, We explain
the connection of this new realization with form factors.
