Projective geometry of Freudenthal's varieties of certain type
type: Item
instance of: scholarly article
title: Projective geometry of Freudenthal's varieties of certain type
zbMATH Open document ID: 1071.14051
author: Hajime Kaji
author: Osami Yasukura
published in: Michigan Mathematical Journal
publication date: 2005-03-07T00:00:00Z
review text: The purpose of this article is to explore the geometry of Freudenthal-type varieties. To obtain such varieties, one begins with a graded Lie algebra \({\mathfrak g}\) (over \({\mathbb C}\)) of contact type, with \[ {\mathfrak g}=\bigoplus _{-2\leq n\leq 2} {\mathfrak g}_n. \] (One may think of this grading as the natural decomposition of \({\mathfrak g}\) under the adjoint action of a copy of \({\mathfrak {sl}}_2\) within \({\mathfrak g}\).) Then, we define the variety \(V_k\) as the projectivization of \(\{x\in {\mathfrak g_1}\mid (\operatorname{ad} x)^{k+1} {\mathfrak g}_{-2}=0\}\). Aspects of the grading of \({\mathfrak g}\) are important in determining the geometry of the \(V_k\). (For example, the Lie bracket \({\mathfrak g}_1 \times {\mathfrak g}_1\rightarrow {\mathfrak g}_2\) induces a symplectic form on \({\mathfrak g}_1\).) Another important result is the relationship between the quartic form on \({\mathfrak g}_1\) and a certain Cremona transformation associated to \({\mathfrak g}_1\). The authors proceed to establish several geometric results, including that \(V_1\) is a homogeneous space; \(V_1\) is a projectivization of a Lagrangian subvariety of \({\mathfrak g}_1\); \(V_1\) is ruled when \(V_2\neq V\); and numerous results about secants and tangents in the projectivization of \(g_1\).
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14N20
Mathematics Subject Classification ID: 14E07
zbMATH DE Number: 2141488
zbMATH Keywords: Cremona transformation
zbMATH Keywords: graded Lie algebra
zbMATH Keywords: secant lines
reviewed by: John Lorch
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1100623411
OpenAlex ID: W1987957685
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DOI: 10.1307/MMJ/1100623411
description: scientific article; zbMATH DE number 2141488