Differential Galois theory I

Differential Galois Theory

The aim of the talk is to explain how to make these statements precise and prove them, using differential Galois theory. The reference to all of the material here is [vdPS03]. By solving a differential equation we mean obtaining a solution via a finite number of operations of the following kind (starting with a rational function): • Adding a function algebraic over the functions we already have...

Differential Galois Theory

Differential Galois Theory is a branch of abstract algebra that studies fields equipped with a derivation function. In much the same way as ordinary Galois Theory studies field extensions generated by solutions of polynomials over a base field, differential Galois Theory studies differential field extensions generated by solutions to differential equations over a base field. In this paper, we w...

Non-linear Differential Galois Theory

Galois algebras I: Structure Theory

We introduce a concept and develop a theory of Galois subalgebras in skew semigroup rings. Proposed approach has a strong impact on the representation theory, first of all the theory of Harish-Chandra modules, of many infinite dimensional algebras including the Generalized Weyl algebras, the universal enveloping algebras of reductive Lie algebras, their quantizations, Yangians etc. In particula...

Algebraic D-groups and Differential Galois Theory

We discuss various relationships between the algebraic Dgroups of Buium, 1992, and differential Galois theory. In the first part we give another exposition of our general differential Galois theory, which is somewhat more explicit than Pillay, 1998, and where generalized logarithmic derivatives on algebraic groups play a central role. In the second part we prove some results with a “constrained...