# Twisted toroidal Lie algebras and Moody-Rao-Yokonuma presentation

## TL;DR

This paper provides an explicit realization and presentation of twisted toroidal Lie algebras associated with affine Kac-Moody algebras and diagram automorphisms, connecting to their quantization.

## Contribution

It introduces a Moody-Rao-Yokonuma presentation for the universal central extension of twisted loop algebras, especially for non-transitive automorphisms.

## Key findings

- Explicit realization of the universal central extension
- Moody-Rao-Yokonuma presentation for non-transitive automorphisms
- Connection to quantization of toroidal Lie algebras

## Abstract

Let $\fg$ be an affine Kac-Moody algebra, and $\mu$ a diagram automorphism of $\fg$. In this paper, we give an explicit realization for the universal central extension $\wh\fg[\mu]$ of the twisted loop algebra of $\fg$ related to $\mu$, which provides a Moody-Rao-Yokonuma presentation for the algebra $\wh\fg[\mu]$ when $\mu$ is non-transitive, and the presentation is indeed related to the quantization of toroidal Lie algebras.