On 𝔰𝔩2-triples for classical algebraic groups in positive characteristic

Simon M. Goodwin; Rachel Pengelly

doi:10.1007/s00031-022-09722-y

On 𝔰𝔩₂-triples for classical algebraic groups in positive characteristic

Simon M. Goodwin, Rachel Pengelly

Mathematics

Research output: Contribution to journal › Article › peer-review

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Abstract

Let 𝕜 be an algebraically closed field of characteristic p > 2, let n ∈ ℤ>0, and take G to be one of the classical algebraic groups GL_n(𝕜), SL_n(𝕜), Sp_n(𝕜), O_n(𝕜) or SO_n(𝕜), with 𝔤 = Lie G. We determine the maximal G-stable closed subvariety V of the nilpotent cone N of 𝔤 such that the G-orbits in V are in bijection with the G-orbits of 𝔰𝔩₂-triples (e, h, f) with e, f ∈ V. This result determines to what extent the theorems of Jacobson–Morozov and Kostant on 𝔰𝔩₂-triples hold for classical algebraic groups over an algebraically closed field of “small” odd characteristic.

Original language	English
Journal	Transformation Groups
DOIs	https://doi.org/10.1007/s00031-022-09722-y
Publication status	Published - 20 May 2022

Bibliographical note

Funding Information:
The results in this paper form part of the PhD research of the second author, who received financial support from the EPSRC grant EP/R513167/1. The first author is supported by EPSRC grant EP/R018952/1.

Publisher Copyright:
© 2022, The Author(s).

Keywords

Algebraic groups
Lie algebras
Positive characteristic
sl -triples

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

Access to Document

10.1007/s00031-022-09722-yLicence: Creative Commons: Attribution (CC BY)

GoodwinS2022Sl2Final published version, 405 KBLicence: Creative Commons: Attribution (CC BY)

Representation theory of modular Lie algebras and superalgebras
Goodwin, S.
Engineering & Physical Science Research Council
1/07/18 → 31/12/22
Project: Research Councils

Cite this

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title = "On 픰픩2-triples for classical algebraic groups in positive characteristic",

abstract = " Let 한 be an algebraically closed field of characteristic p > 2, let n ∈ ℤ>0, and take G to be one of the classical algebraic groups GLn(한), SLn(한), Spn(한), On(한) or SOn(한), with 픤 = Lie G. We determine the maximal G-stable closed subvariety V of the nilpotent cone N of 픤 such that the G-orbits in V are in bijection with the G-orbits of 픰픩2-triples (e, h, f) with e, f ∈ V. This result determines to what extent the theorems of Jacobson–Morozov and Kostant on 픰픩2-triples hold for classical algebraic groups over an algebraically closed field of “small” odd characteristic. ",

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