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Journal of Lie Theory 28 (2018), No. 1, 057--070
Copyright Heldermann Verlag 2018

Varieties of Elementary Subalgebras of Submaximal Rank in Type A

Yang Pan
School of Sciences, Zhejiang A&F University, Huanbei Road 88, 311300 Hangzhou, P.R.China
and: Mathematisches Seminar, Universitšt Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany


\def\g{{\frak g}} \def\E{\mathbb E} Let $G$ be a connected simple algebraic group over an algebraically closed field {\bf k} of characteristic $p>0$, and $\g$ = lie$(G)$. We additionally assume that $G$ is standard and is of type $A_{n}$. Motivated by the investigation of the geometric properties of the varieties $\E(r,\g)$ of $r$-dimensional elementary subalgebras of a restricted Lie algebra $\g$, we will show in this article the irreducible components of $\E({\rm rk}_p(\g)-1,\g)$ when rk$_p(\g)$ is the maximal dimension of an elementary subalgebra of $\g$.

Keywords: Elementary subalgebras, irreducible components.

MSC: 17B50, 16G10

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