Drinfeld-type presentations of loop algebras

Abstract

Let g \mathfrak {g} be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let μ \mu be a diagram automorphism of g \mathfrak {g} , and let L ( g , μ ) \mathcal {L}(\mathfrak {g},\mu ) be the loop algebra of g \mathfrak {g} associated to μ \mu . In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension g ^ [ μ ] \widehat {\mathfrak {g}}[\mu ] of L ( g , μ ) \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 g \mathfrak {g} is of simply-laced-type, we prove that the classical limit of the μ \mu -twisted quantum affinization of the quantum Kac-Moody algebra associated to g \mathfrak {g} introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of g ^ [ μ ] \widehat {\mathfrak {g}}[\mu ] .