Normal forms in differential Galois theory for the classical groups

Communications in Algebra, vol. 51, pp. 1492–1516

Abstract

Let G be a classical group of dimension d and let a=(a1,…,ad) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A(a) be a generic element in the Lie algebra g(F〈a〉) of G obtained from parametrizing a basis of g with the indeterminates a. It is known (cf. [5 Juan, L. (2004). Pure Picard-Vessiot extensions with generic properties. Proc. Amer. Math. Soc. 132(9):2549–2556. DOI: 10.1090/S0002-9939-04-07390-3.[Crossref], [Web of Science ®] , [Google Scholar]]) that the differential Galois group of y′=A(a)y over F〈a〉 is G(C). In this paper we construct a differential field extension L of F〈a〉 such that the field of constants of L is C, the differential Galois group of y′=A(a)y over L is still the full group G(C) and A(a) is gauge equivalent over L to a matrix in normal form which we introduced in [13 Seiß, M. (2016). Root parametrized differential equations for the classical groups. arXiv:1609.05535. [Google Scholar]]. We also consider specializations of the coefficients of A(a).

Authors 2

1. Daniel Robertz Aachen Lehrstuhl für Algebra und Zahlentheorie

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, Aachen, Germany

2. Matthias Seiß corresponding

   University of Kassel

   Affiliation as printed

   Institut für Mathematik, Universität Kassel, Kassel, Germany