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On Varieties of Lie Algebras of Maximal Class

Published:2014-04-28

• Tatyana Barron,
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7
• Dmitry Kerner,
Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel
Fields Institute for Research in Mathematical Sciences, Toronto, ON, M5T 3J1
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Abstract

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 A. Fialowski there are only three isomorphism types of $\mathbb{N}$-graded Lie algebras $L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$ and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the structure of these algebras with the property $L=\langle L_1 \rangle$. In this paper we study those generated by the first and $q$-th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. 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.
 Keywords: filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification
 MSC Classifications: 17B70 - Graded Lie (super)algebras 14F45 - Topological properties