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=∞i=1 Li of maximal class generated by L1 and L2, L=<L1,L2>. Vergne described the structure of these algebras with the property L=<L1>. In this paper we study those generated by the first and q-th components where q>2, L=<L1,Lq>. 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.
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