On Varieties of Lie Algebras of Maximal Class

We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over $\mathbb{C}$ using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on $\mathbb{N}$ -graded Lie algebras of maximal class. As shown by $\text{A}$ . Fialowski there are only three isomorphism types of $\mathbb{N}$ -graded Lie algebras $L\,=\,\oplus _{i=1}^{\infty }\,{{L}_{i}}$ of maximal class generated by ${{L}_{i}}$ and ${{L}_{2}}$ , $L\,=\,\left\langle {{L}_{1}},\,{{L}_{2}} \right\rangle$ . Vergne described the structure of these algebras with the property $L\,=\,\left\langle {{L}_{1}} \right\rangle$ . In this paper we study those generated by the first and $q$ -th components where $q\,>\,2$ , $L\,=\,\left\langle {{L}_{1}},\,{{L}_{q}} \right\rangle$ . 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.