Partial Gauss decomposition, bf U_q(widehat{{{frak{gl}}}(n-1)})in U_q(widehat{{{frak{gl}}}(n)}) and Zamolodchikov algebra

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.