Automorphisms and twisted forms of the \(N = 1, 2, 3\) Lie conformal superalgebras
type: Item
instance of: scholarly article
title: Automorphisms and twisted forms of the \(N = 1, 2, 3\) Lie conformal superalgebras
zbMATH Open document ID: 1271.17020
author: Zhihua Chang
author: Arturo Pianzola
published in: Communications in Number Theory and Physics
publication date: 2012-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.5410
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 17B81
Mathematics Subject Classification ID: 81R10
Mathematics Subject Classification ID: 14D21
zbMATH DE Number: 6092505
zbMATH Keywords: Superconformal Lie algebras
zbMATH Keywords: conformal Lie superalgebras
MaRDI profile type: Publication
arXiv ID: 1201.5410
OpenAlex ID: W2963438522
DOI: 10.4310/CNTP.2011.V5.N4.A1
review text: The authors have a twofold purpose with the article. Firstly, to give the physicists new ideas, that is methods coming from differential non-abelian cohomology. These methods are applied on reductive group schemes, torsors, and descent, and can be adapted to the study of Lie conformal superalgebras by replacing the base scheme \(\text{Spec}(\mathbb C[t^{\pm 1}]\) in this case, by a differential scheme.NEWLINENEWLINEThe second purpose is to complete the classification of the \(N=1,2,3,4\) Lie superalgebras, referred to as superconformal Lie algebras by the physicists. The authors give a uniform proof of the cases \(N=1,2,3\), by using the fact that the relevant Lie conformal superalgebras used as base objects for the twisted loop construction can be described in terms of exterior algebras. Also, a precise argument that describes the passage from Lie conformal superalgebras to their corresponding Lie superalgebras is given.NEWLINENEWLINEThe \(N=1,2,3,4\) Lie superalgebras are closely related to the twisted loop Lie conformal superalgebra based on a complex Lie conformal superalgebra. They were realized as differential Lie conformal algebras. A twisted loop Lie conformal superalgebra has both a complex conformal superalgebra structure and a \(\mathcal R\)-Lie conformal superalgebra structure, where \(\mathcal R=(\mathbb C[t^{\pm 1}],\frac{d}{dt}).\) The \(\mathcal R\)-structure makes it possible to introduce cohomological methods.NEWLINENEWLINENEWLINEKac-Moody Lie algebras are defined a priori by generators and relations, and the construction of the twisted loop Lie conformal superalgebras is built on the same principles. The classification of these has been established by non-abelian étale cohomology. The loop algebras based on a finite-dimensional simple Lie algebra \(\mathfrak g\) are parameterized by the conjugacy classes of the finite group of symmetries of the corresponding Coxeter-Dynkin diagram. If the automorphism group of \(\mathfrak g\) is connected, then al…
description: scientific article; zbMATH DE number 6092505