On Picard-Vessiot extensions with group PGL3

Lourdes Juan; Arne Ledet

doi:10.1016/j.jalgebra.2007.09.022

On Picard-Vessiot extensions with group PGL₃

Lourdes Juan, Arne Ledet

Mathematics and Statistics

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Abstract

Let F be a differential field of characteristic zero with algebraically closed field of constants. We provide an explicit description of the twisted Lie algebras of PGL₃-equivariant derivations on the coordinate rings of F-irreducible PGL₃-torsors in terms of nine-dimensional central simple algebras over F. We use this to construct a Picard-Vessiot extension which is the function field of a non-trivial torsor and which is a generic extension for PGL₃.

Original language	English
Pages (from-to)	786-793
Number of pages	8
Journal	Journal of Algebra
Volume	318
Issue number	2
DOIs	https://doi.org/10.1016/j.jalgebra.2007.09.022
State	Published - Dec 15 2007

Keywords

Differential Galois theory
Galois cohomology
Generic extensions
Picard-Vessiot

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10.1016/j.jalgebra.2007.09.022

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abstract = "Let F be a differential field of characteristic zero with algebraically closed field of constants. We provide an explicit description of the twisted Lie algebras of PGL3-equivariant derivations on the coordinate rings of F-irreducible PGL3-torsors in terms of nine-dimensional central simple algebras over F. We use this to construct a Picard-Vessiot extension which is the function field of a non-trivial torsor and which is a generic extension for PGL3.",

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AB - Let F be a differential field of characteristic zero with algebraically closed field of constants. We provide an explicit description of the twisted Lie algebras of PGL3-equivariant derivations on the coordinate rings of F-irreducible PGL3-torsors in terms of nine-dimensional central simple algebras over F. We use this to construct a Picard-Vessiot extension which is the function field of a non-trivial torsor and which is a generic extension for PGL3.

KW - Differential Galois theory

KW - Galois cohomology

KW - Generic extensions

KW - Picard-Vessiot

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On Picard-Vessiot extensions with group PGL₃

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Keywords

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