Automorphisms of finite order of the affine Lie algebra \(A_{\ell}^{(1)}\)
type: Item
instance of: scholarly article
title: Automorphisms of finite order of the affine Lie algebra \(A_{\ell}^{(1)}\)
zbMATH Open document ID: 0618.17008
author: Zenji Kobayashi
published in: Tsukuba Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathfrak g}\) be the affine Lie algebra over \({\mathbb{C}}\) of type \(A_{n- 1}(1)\) (for \(n\geq 2)\). The author explicitly determines all the automorphisms of prime order of the Lie algebra \({\mathfrak g}\) upto conjugacy in Aut(\({\mathfrak g})\). By the universality of the central extension \[ 0 \to {\mathbb{C}}Z \to {\mathfrak g} \to s\ell_ n({\mathbb{C}}[t,t^{- 1}]) \to 0, \] it can be seen that Aut(\({\mathfrak g})\approx Aut_{{\mathbb{C}}}(s\ell_ n({\mathbb{C}}[t,t^{-1}]))\). Now the author reduces the study of elements of prime order k in \(Aut_{{\mathbb{C}}}(s\ell_ n({\mathbb{C}}[t,t^{-1}]))\) (upto conjugacy) to the study of the group cohomology \(H^ 1({\mathbb{Z}}_ k, PGL_ n({\mathbb{C}}[t,t^{-1}]))\) with \({\mathbb{Z}}_ k\) acting (in general) nontrivially on \(PGL_ n\). The large part of the paper is devoted to determine \(H^ 1({\mathbb{Z}}_ k,PGL_ n({\mathbb{C}}[t,t^{-1}]))\) explicitly, which is done case by case and which uses Grothendieck's ''theory of descent''.
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 4002338
zbMATH Keywords: nonabelian group cohomology
zbMATH Keywords: affine Lie algebra
zbMATH Keywords: automorphisms of prime order
Wikidata QID: Q115233618
reviewed by: Shrawan Kumar
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160457
OpenAlex ID: W1920416108
DOI: 10.21099/TKBJM/1496160457
description: scientific article; zbMATH DE number 4002338