On varieties of Lie algebras of maximal class

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 (see also [shalev:97], [millionshchikov:04]) there are only three isomorphism types of N-graded Lie algebras $L=\oplus^{\infty}_{i=1} L_i$ 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.