Related papers: Lie's Reduction Method and Differential Galois The…
Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides…
Let $\mathbf{k}$ be a differential field and let $[A]\,:\,Y'=A\,Y$ be a linear differential system where $A\in\mathrm{Mat}(n\,,\,\mathbf{k})$. We say that $A$ is in a reduced form if $A\in\mathfrak{g}(\bar{\mathbf{k}})$ where $\mathfrak{g}$…
Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, many…
We show how the Galois-Picard_Vessiot theory of differential equations and difference equations, and the theory of holonomy groups in differential geometry, are different aspects of a unique Galois theory. The latter is based upon the…
We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three…
In this paper, we describe Galois covers of algebraic curves and their families by using local systems associated to push-forward of sheaves by the structure morphism. More precisely, if $f:C\to Y$, we consider the sheaves $f_*(\C)$. The…
The Galois theory of logarithmic differential equations with respect to relative D-groups in partial differential-algebraic geometry is developed.
Over a smooth and proper complex scheme, the differential Galois group of an integrable connection may be obtained as the closure of the transcendental monodromy representation. In this paper, we employ a completely algebraic variation of…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
We define and analyse the properties of contact Lie systems, namely systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of…
We develop algorithms to compute the differential Galois group corresponding to a one-parameter family of second order homogeneous ordinary linear differential equations with rational function coefficients. More precisely, we consider…
A new singular perturbation method based on the Lie symmetry group is presented to a system of difference equations. This method yields consistent derivation of a renormalization group equation which gives an asymptotic solution of the…
We prove the Galois correspondence between the subgroups of a finite automorphism group G of a simple vertex operator algebra V and the vertex operator subalgebras of V containing the set V^G of G-invariants.
The automorphism group of the Galois covering induced by a pluri-canonical generic covering of a projective space is investigated. It is shown that by means of such coverings one obtains, in dimensions one and two, serieses of specific…
We explore the relationship between the category of MV-algebras and its full subcategories of perfect and semisimple algebras, showing that this pair of subcategories defines a pretorsion theory. We study the Galois structure associated…
This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat…
After reviewing the basic role of Lie's theory for the mathematics and physics of this century, we identify its limitations for the treatment of systems beyond the local-differential, Hamiltonian and canonical-unitary conditions of the…
In this paper we extend the Lie theory of integration in two different ways. First we consider a finite dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields…
In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and…
We develop a Galois theory for difference ring extensions, inspired by Magid's separable Galois theory for ring extensions and by Janelidze's categorical Galois theory. Our difference Galois theorem states that the category of difference…