Secant varieties of adjoint varieties: Orbit decomposition
type: Item
instance of: scholarly article
title: Secant varieties of adjoint varieties: Orbit decomposition
zbMATH Open document ID: 0986.14034
author: Hajime Kaji
author: Osami Yasukura
published in: Journal of Algebra
publication date: 2001-10-29T00:00:00Z
review text: Let \(\mathfrak g\) be a complex simple Lie algebra. By the adjoint variety \(X(\mathfrak{g})\) the authors mean the unique closed orbit of the action of \(G\) (the inner automorphism group of \(\mathfrak g\)) on \({\mathfrak P}(\mathfrak g) \) induced by the adjoint representation of \(G\) on \(\mathfrak g\). Adjoint varieties are interesting because their secant varieties have \textit{defect}, that is, their dimension is less than the generic dimension. This was discussed in a previous paper by the first author [\textit{H. Kaji}, Math. Contemp. 14, 75-87 (1998; Zbl 0939.14030)]. The action of \(G\) on \(X(\mathfrak g)\) lifts to an action on the secant variety of \(X(\mathfrak g)\), and the purpose of the present paper is to give the orbit decomposition of this action for the different complex simple Lie algebras. A structure theorem is proved, which states that the secant variety decomposes as the union of a dense orbit and a finite number of projectivizations of nilpotent orbits. The nilpotent orbits have a unique maximal orbit relative to the closure ordering. A complete classification of these maximal orbits is given.
Mathematics Subject Classification ID: 14N30
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14N15
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 17B40
zbMATH DE Number: 1448636
zbMATH Keywords: orbit decomposition
zbMATH Keywords: adjoint representation
zbMATH Keywords: secant variety
zbMATH Keywords: nilpotent orbits
zbMATH Keywords: maximal orbit
reviewed by: T. G. Berry
MaRDI profile type: Publication
full work available at URL: jabr.1999.8223
OpenAlex ID: W2073751480
cites work: Classes of unipotent elements in simple algebraic groups. I
cites work: Classes of unipotent elements in simple algebraic groups. II
cites work: Fano contact manifolds and nilpotent orbits
cites work: Q3287750
cites work: A Note on Homogeneous Complex Contact Manifolds
cites work: Q5580319
cites work: Q4376915
cites work: Q4863383
cites work: Lie groups in the foundations of geometry
cites work: Varieties with Small Secant Varieties: The Extremal Case
cites work: Q4002278
cites work: Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili
cites work: Q4143433
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q4744461
cites work: A Classification of Spinors Up to Dimension Twelve
cites work: Adjoint varieties and their secant varieties
cites work: Projective geometry of Freudenthal's varieties of certain type
cites work: Q3788656
cites work: Q4942223
cites work: Theb-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces
cites work: Q3677859
cites work: A classification of irreducible prehomogeneous vector spaces and their relative invariants
cites work: Linear algebraic groups
cites work: Q5662587
cites work: THE WEYL GROUP OF A GRADED LIE ALGEBRA
cites work: Q4106713
cites work: Q3140224
DOI: 10.1006/JABR.1999.8223
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description: scientific article; zbMATH DE number 1448636