arXiv subjects

David Blázquez-Sanz

Publications and source records attributed to David Blázquez-Sanz.

22 records · Page 2Linked to original sources

arXiv2009

Local and Global Aspects of Lie's Superposition Theorem

In this paper we give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the well-known Lie superposition theorem. We introduce rigorous notions of pretransitive Lie group action and Lie-Vessiot systems. We proof that pretransitive Lie group actions are transitive. We proof that an ordinary differential equation admit a superposition law if and only if it is a pretransitive Lie-Vessiot system. It means that its enveloping algebra is spanned by fundamental fields of a pretransitive Lie group action. We discuss the relationship of superposition laws with differential Galois theory and review the classical result of Lie.

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arXiv2009

Lie's Reduction Method and Differential Galois Theory in the Complex Analytic Context

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generaliza- tion of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot sys- tem is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. Lie. We propose an differential Galois theory for automorphic systems, and explore the relationship between integrability in terms of Galois the- ory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system.

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arXiv2009

Differential Galois Theory of Algebraic Lie-Vessiot Systems

In this paper we develop a differential Galois theory for algebraic Lie-Vessiot systems in algebraic homogeneous spaces. Lie-Vessiot systems are non autonomous vector fields that are linear combinations with time-dependent coefficients of fundamental vector fields of an algebraic Lie group action. Those systems are the building blocks for differential equations that admit superposition of solutions. Lie-Vessiot systems in algebraic homogeneous spaces include the case of linear differential equations. Therefore, the differential Galois theory for Lie-Vessiot systems is an extension of the classical Picard-Vessiot theory. In particular, algebraic Lie-Vessiot systems are solvable in terms of Kolchin's strongly normal extensions. Therefore, strongly normal extensions are geometrically interpreted as the fields of functions on principal homogeneous spaces over the Galois group. Finally we consider the problem of integrability and solvability of automorphic differential equations. Our main tool is a classical method of reduction, somewhere cited as Lie reduction. We develop and algebraic version of this method, that we call Lie-Kolchin reduction. Obstructions to the application are related to Galois cohomology.

math.CA↗

arXiv2009

Affine Structures on Jet and Weil Bundles

Weil algebra morphism induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cases, this structure of affine bundle is passed down to Jet spaces. We give a characterization of this fact in algebraic terms. This algebraic condition also determines an affine structure between the groups of automorphisms of related Weil algebras

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