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, let $\mu$ be a diagram automorphism of $\mathfrak {g}$, and let $\mathcal {L}(\mathfrak {g},\mu )$ be 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 $\mathcal {L}(\mathfrak {g},\mu )$. The construction contains the classical limit of Drinfeld’s new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212–216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283–307] 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 [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of $\widehat {\mathfrak {g}}[\mu ]$.