Structure of basic Lie superalgebras and of their affine extensions
type: Item
instance of: scholarly article
title: Structure of basic Lie superalgebras and of their affine extensions
zbMATH Open document ID: 0671.17014
DOI: 10.1007/BF01217734
published in: Communications in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: An affine Kac-Moody algebra is the central extension of a (twisted) loop algebra. Twists are associated with automorphisms of the Dynkin diagram of the finite-dimensional target algebra. The reviewer's conjecture was that the same is true for superalgebras. V. Serganova listed all outer automorphisms and the associated (twisted) loop superalgebras providing with fast experimental material to back up this conjecture and correct it: nonisomorphic twisted superalgebras correspond to automorphisms modulo the connected component of the unit of the group of automorphisms, which for superalgebras may be wider than the group of inner automorphisms. For superalgebras with symmetrizable Cartan matrix this conjecture had been proved by J. van de Leur. In order to list all the systems of simple roots for the above Lie superalgebras (a Lie superalgebra may possess several different ones and previously there were numerous partial results) V. Serganova introduced several interesting analogues of Weyl group. The authors dutifully list all the relevant bibliography and expose some of Serganova's results. In particular they show that with an automorphism of the Dynkin diagram of a Lie superalgebra (finite-dimensional or loopy) a new algebra is associated. It is shown that whereas all simple finite- dimensional Lie algebras can be obtained this way from A-D-E diagrams, for superalgebras the applicability of this method is more restricted. From Serganova's results one easily derives a description of maximal regular subalgebras of finite-dimensional algebras (this has been independently performed by J. Van der Jeugt with some omissions corrected in the text under review).
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17A70
zbMATH DE Number: 4097598
zbMATH Keywords: affine Kac-Moody algebra
zbMATH Keywords: loop algebra
zbMATH Keywords: outer automorphisms
zbMATH Keywords: loop superalgebras
zbMATH Keywords: twisted superalgebras
zbMATH Keywords: automorphism
zbMATH Keywords: Dynkin diagram
zbMATH Keywords: Lie superalgebra
zbMATH Keywords: maximal regular subalgebras
author: L. Frappat
author: A. Sciarrino
author: P. Sorba
reviewed by: Dimitry Leites
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4097598