Local coordinate systems on quantum flag manifolds

TL;DR

This paper explores quantum flag manifolds by applying Feigin's homomorphisms to establish fundamental theorems, and introduces invariants and generators for lattice Virasoro algebras related to quantum affine Lie algebras.

Contribution

It provides new results on generators of lattice Virasoro algebras and applies Feigin's homomorphisms to quantum Serre relations and screening operators.

Findings

Found two and three point invariants for $U_q( ilde{sl}_2)$

Identified generators of lattice Virasoro algebra for $sl_2$

Established fundamental theorems related to quantum Serre relations

Abstract

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.