Classification des algèbres de Lie graduées simples de croissance \(\leq 1\). (Classification of simple graded Lie algebras of growth \(\leq 1)\)
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 dimension of to an affine Kac-Moody Lie algebra (resp. to W or \(W_ 1).\) In the 1984 paper, the author proved that if \(g=\oplus g_ n\) is a simple graded Lie algebra of growth \(\leq 1\) then a Cartan subalgebra of \(g_ 0\) has a root. Here, he conjectures that the same holds if ''of growth \(\leq 1''\) is replaced by ''of finite growth''. He then proves that it is a special case of Kac's conjecture. In a note added in proof the author announces that he succeeded in proving his conjecture.
scientific article; zbMATH DE number 3995932