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arXiv · 2101.11982

Thin subalgebras of Lie algebras of maximal class

M. Avitabile·A. Caranti·N. Gavioli·V. Monti·M. F. Newman·E. A. O'Brien
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Abstract

For every field $F$ which has a quadratic extension $E$ we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension $2$. We construct such Lie algebras as $F$-subalgebras of Lie algebras $M$ of maximal class over $E$. We characterise the thin Lie $F$-subalgebras of $M$ generated in degree $1$. Moreover we show that every thin Lie algebra $L$ whose ring of graded endomorphisms of degree zero of $L^3$ is a quadratic extension of $F$ can be obtained in this Lie algebra of maximal class over $E$ which are ideally $r$-constrained for a positive integer $r$.

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M. Avitabile, A. Caranti, N. Gavioli, V. Monti, M. F. Newman, E. A. O'Brien. 2021-01-28. Thin subalgebras of Lie algebras of maximal class. https://arxiv.org/abs/2101.11982

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