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 $\hat{\mathfrak{g}}[\mu ]$ 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 $\hat{\mathfrak{g}}[\mu ]$.