Drinfeld type presentations of loop algebras

Abstract

Let $\\mathfrak{g}$ be the derived subalgebra of a Kac-Moody Lie algebra of finite type or affine type, $\\mu$ a diagram automorphism of $\\mathfrak{g}$ and $L(\\mathfrak{g},\\mu)$ the loop algebra of $\\mathfrak{g}$ associated to $\\mu$. In this paper, by using the vertex algebra technique, we provide a general construction of current type presentations for the universal central extension $\\widehat{\\mathfrak{g}}[\\mu]$ of $L(\\mathfrak{g},\\mu)$. The construction contains the classical limit of Drinfeld s new realization for (twisted and untwisted) quantum affine algebras ([Dr]) and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras ([MRY]) as special examples. As an application, when $\\mathfrak{g}$ is of simply-laced type, we prove that the classical limit of the $\\mu$-twisted quantum affinization of the quantum Kac-Moody algebra associated to $\\mathfrak{g}$ introduced in [CJKT1] is the universal enveloping algebra of $\\widehat{\\mathfrak{g}}[\\mu]$.