Hopf Algebraic Approach to Picard-Vessiot Theory

DOI10.1016/S1570-7954(08)00204-0zbMath1218.12003OpenAlexW1877924354MaRDI QIDQ3053861

Akira Masuoka, Katsutoshi Amano, Mitsuhiro Takeuchi

Publication date: 30 October 2010

Published in: Handbook of Algebra - Volume 6 (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/s1570-7954(08)00204-0

zbMATH Keywords

Hopf algebradifferential Galois theoryabstract differential equationsPicard-Vessiot theory

Mathematics Subject Classification ID

Differential algebra (12H05) Abstract differential equations (12H20) Hopf algebras and their applications (16T05) Research exposition (monographs, survey articles) pertaining to field theory (12-02)

Related Items (14)

Twisted forms of differential Lie algebras over \(\mathbb{C}(t)\) associated with complex simple Lie algebras ⋮ The differential Galois group of the rational function field ⋮ Toward quantization of Galois theory ⋮ Parameterized Picard-Vessiot extensions and Atiyah extensions ⋮ Existence of \(\partial\)-parameterized Picard-Vessiot extensions over fields with algebraically closed constants ⋮ An intrinsic characterization of Picard-Vessiot extensions ⋮ Difference Galois theory of linear differential equations ⋮ On the finite dual of a cocommutative Hopf algebroid. Application to linear differential matrix equations and Picard-Vessiot theory. ⋮ Algebraic groups as difference Galois groups of linear differential equations ⋮ Differential embedding problems over complex function fields ⋮ Generalized \(q\)-boson algebras and their integrable modules ⋮ Generalized differentials and prolongation spaces ⋮ Free differential Galois groups ⋮ Galois Theory of Difference Equations with Periodic Parameters

Cites Work

• Differential Galois theory and tensor products
• Differential function fields and moduli of algebraic varieties
• A Hopf algebraic approach to the Picard-Vessiot theory
• Hopf-Galois extensions of algebras, the Miyashita-Ulbrich action, and Azumaya algebras
• Galois theory of difference equations
• Picard-Vessiot extensions of Artinian simple module algebras.
• Basic properties of differential fields of an arbitrary characteristic and the Picard-Vessiot theory
• Affine Hopf algebras. II
• A correspondence between Hopf ideals and sub-Hopf algebras
• The Predual Theorem to the Jacobson-Bourbaki Theorem
• Différentielles non commutatives et théorie de Galois différentielle ou aux différences
• Liouville Extensions of Artinian Simple Module Algebras
• Picard-Vessiot Theory of Linear Homogeneous Difference Equations
• Picard-Vessiot Theory of Partial Differential Fields
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