Year: 2002
Type: article
Abstract: The analogs of Chevalley generators are offered for simple (and close to them) Z-graded complex Lie algebras and Lie superalgebras of polynomial growth without Cartan matrix.We show how to derive the defining relations between these generators and explicitly write them for a "most natural" ("distinguished" in terms of Penkov and Serganova) system of simple roots.The results are given mainly for Lie superalgebras whose component of degree zero is a Lie algebra (other cases being left to the reader).Observe presentations presentations of exceptional Lie superalgebras and Lie superalgebras of hamiltonian vector fields. To Jan-Erik Roos on his sixty-fifth birthday §1. PreliminariesAfter Berezin formulated to one of us (DL) the problem which in modern terms would sound "define supermanifolds and differential supergeometry" it was natural to look after examples of Lie superalgebras that naturally appear in mathematics.Homotopy rings with respect to Whitehead's product are examples of such Lie rings, but nobody, so far, described these natural rings in any case.By description we mean identification of the semisimple part and radical.A reason for such careless attention to these rings becomes clear soon after one tackles the problem: they are nilpotent, hence, not so interesting in a sense (simple Lie (super)algebras have a richer structure and interesting representation theory).A paper by C. Löfvall and J.-E.Roos [LR] is a break-through: in a similar problem they made a very interesting observation: they not only found traces left by simple Lie superalgebras where nothing indicated them, they also identified these superalgebras as a "positive part" of certain twisted loops algebras with values in simple Lie superalgebras.The paper [LR] is, clearly, the first in a series to appear, We were financially supported: P.G. by the Swedish Institute in 1993-95; D.L. by I. Bendixson grant, an
Authors: P. Grozman, D. Leites, E. Poletaeva
Institutions: Stockholm University, Lund University
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Funder: National Science Foundation
OpenAlex ID: W2105054434