Local coordinate systems on quantum flag manifolds

This paper consist of 3 sections. In the first section, we will give a brief introduction to the "Feigin's homomorphisms" and will see how they will help us to prove our main and fundamental theorems related to quantum Serre relations and screening operators. In the second section, we will introduce Local integral of motions as the space of invariants of nilpotent part of quantum affine Lie algebras and will find two and three point invariants in the case of $U_q(\hat{sl_2})$ by using Volkov's scheme. In the third section, we will introduce lattice Virasoro algebras as the space of invariants of Borel part $U_q(B_{+})$ of $U_q(g)$ for simple Lie algebra $g$ and will find the set of generators of Lattice Virasoro algebra connected to $sl_2$ and $U_q(sl_2)$. And as a new result, we found the set of some generators of lattice Virasoro algebra.