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arXiv · 1207.5898

Elementary subalgebras of Lie algebras

Jon F. Carlson·Eric M. Friedlander·Julia Pevtsova
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Abstract

We initiate the investigation of the projective variety $E(r,g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $g$ for some $r > 0$ and demonstrate that this variety encodes considerable information about the representations of $g$. For various choices of $g$ and $r$, we identify the geometric structure of $E(r,g)$. We show that special classes of (restricted) representations of $g$ lead to algebraic vector bundles on $E(r,g)$. For $g = Lie(G)$ the Lie algebra of an algebraic group $G$, rational representations of $G$ enable us to realize familiar algebraic vector bundles on $G$-orbits of $E(r, g)$.

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Jon F. Carlson, Eric M. Friedlander, Julia Pevtsova. 2012-07-25. Elementary subalgebras of Lie algebras. https://arxiv.org/abs/1207.5898

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