Toward a vertex operator construction of quantum affine algebras

We describe a construction of the quantum-deformed affine algebras using vertex operators in the free field theory. We prove the Serre relations for the Borel subalgebras of quantum affine algebras; in particular, we consider the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\widehat{sl}_2 $$ \end{document} case in detail. We also construct the generators corresponding to the positive roots of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\widehat{sl}_2 $$ \end{document}.