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

References in corpus (7)

• Braid Group Action and Quantum Affine Algebras
• From Dynkin diagram symmetries to fixed point structures
• $\Gamma$-Conformal Algebras
• PBW and duality theorems for quantum groups and quantum current algebras
• Twisted toroidal Lie algebras and Moody-Rao-Yokonuma presentation
• Twisted quantum affinizations and their vertex representations
• Universal central extensions of twisted forms of split simple Lie algebras over rings

Cited by in corpus (1)

• Twisted quantum affinizations and quantization of extended affine Lie algebras