Classification des algèbres de Lie graduées simples de croissance \(\leq 1\). (Classification of simple graded Lie algebras of growth \(\leq 1)\)
type: Item
instance of: scholarly article
title: Classification des algèbres de Lie graduées simples de croissance \(\leq 1\). (Classification of simple graded Lie algebras of growth \(\leq 1)\)
zbMATH Open document ID: 0615.17010
DOI: 10.1007/BF01389076
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143400
review text: In 1968, \textit{V. G. Kac} [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1323-1367 (1968), translation in Math. USSR, Izv. 2, 1271-1311 (1968; Zbl 0222.17007)] proved that a simple graded Lie algebra \(g=\oplus g_ n\) of finite growth is isomorphic to an affine Kac-Moody Lie algebra or to a Cartan Lie algebra (i.e. some remarkable subalgebra of Der K[T\({}_ 1,...,T_ n])\) iff: 1) g is infinite-dimensional, 2) \(g_{-1}\oplus g_ 0\oplus g_ 1\) generates g, 3) the \(g_ 0\)-module \(g_{-1}\) is simple and faithful. He then conjectured that if one drops these 3 conditions, one gets only 2 extra cases: a simple Lie algebra of finite- dimension and the centerless Virasoro algebra \(W=Der K[T,T^{-1}].\) In 1982, \textit{I. Kaplansky} [Commun. Math. Phys. 86, 49-54 (1982; Zbl 0515.17007)] proved that a graded Lie algebra \(g=\oplus g_ n\) is isomorphic to W iff 1) dim \(g_ n=1\) for all n, 2) \(g_{-1}\oplus g_ 0\oplus g_ 1\cong sl(2)\), 3) \([g_{-1},g_ 2]\neq (0)\), 4) \([g_ 1,g_{-2}]\neq (0).\) In 1984, the author [Preprint of the Univ. Paris VII, 1984; published in J. Algebra 102, 505-536 (1986; Zbl 0596.17012)] proved that a simple graded algebra \(g=\oplus g_ n\) such that dim \(g_ n\leq 1\) for all n is isomorphic to sl(2), W, \(W_ 1\) or to the affine Kac-Moody Lie algebras \(A_ 1^{(1)}\), \(A_ 2^{(2)}.\) In this work the author now gets closer to the conjecture of Kac by proving that a simple graded Lie algebra of growth \(\leq 1\) is isomorphic to a finite-dimensional simple Lie algebra, to an affine Kac-Moody Lie algebra, to W or to \(W_ 1.\) The proof is very long and extremely intricated (it needs 59 non trivial lemmas) and one wonders how the author found his way out of this labyrinth. The notion of integrable algebra is very important (it corresponds more or less to the local nilpotency of some elements). Theorem 3 (resp. theorem 2) says that a simple graded (resp. non-) integrable Lie algebra of growth \(\leq 1\) is isomorphic to a simple Lie algebra of finite dimensio…
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 3995932
zbMATH Keywords: Virasoro algebra
zbMATH Keywords: simple graded Lie algebra
zbMATH Keywords: growth
zbMATH Keywords: affine Kac-Moody Lie algebra
zbMATH Keywords: integrable Lie algebra
Wikidata QID: Q106358848
author: Olivier Mathieu
MaRDI profile type: Publication
cites work: Q4095018
cites work: Q4065692
cites work: On defining relations of certain infinite-dimensional Lie algebras
cites work: Q3892400
cites work: SIMPLE IRREDUCIBLE GRADED LIE ALGEBRAS OF FINITE GROWTH
cites work: Q3955548
cites work: Infinite dimensional Lie algebras. An introduction
cites work: The Virasoro algebra
cites work: Q5541400
description: scientific article; zbMATH DE number 3995932