On Varieties of Lie Algebras of Maximal Class

Abstract

Abstract We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over ℂ, using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on ℕ-graded Lie algebras of maximal class. As shown by A. Fialowski there are only three isomorphism types of ℕ-graded Lie algebras of maximal class generated by L 1 and L 2 , L = 〈 L 1 ; L 2 〉. Vergne described the structure of these algebras with the property L = 〈 L 1 〉. In this paper we study those generated by the first and q-th components where q > 2, L = 〈 L 1 ; L q 〉. Under some technical condition, there can only be one isomorphism type of such algebras. For q = 3 we fully classify them. This gives a partial answer to a question posed by Millionshchikov.