Extensions of differential representations ofSL₂and tori

Abstract

Linear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with SL 2 and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of SL 2 . In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.

Reductive Linear Differential Algebraic Groups and the Galois Groups of Parameterized Linear Differential Equations

“…This result completes and substantially extends what could be proved using [40], where one is restricted just to SL 2 , one derivation, and to those representations that are extensions of just two irreducible representations. We expect that the main results of the present paper will be used in the future to give a complete classification of differential representations of semisimple LDAGs (as this was partially done for SL 2 in [40]). Although reductive and semisimple differential algebraic groups were studied in [13,39], the techniques used there were not developed enough to achieve the goals of this paper.…”

“…Initial steps to understand representations of LDAGs are given in [8,9] and a classification of semisimple LDAGs is given in [13]. A Tannakian approach to the representation theory of LDAGs was introduced in [44,45] (see also [29,28]) and successfully used to further our understanding of representations of reductive LDAGs in [39,40]. This Tannakian approach gives a powerful tool in which one can understand the impact of taking derivatives on the representation theory of LDAGs.…”

Parameterized Picard–Vessiot extensions and Atiyah extensions

“…This assumption is satisfied by many differential fields used in practice, Theorem 2.8.The importance of the existence of a PPV extension is that it leads to a Galois correspondence, Section 8.1. The Galois group is a differential algebraic group [8,9,44,62,11,55,56,71] defined over the field of constants, which, after passing to the differential closure, coincides with the parameterized differential Galois group from [10], Corollary 8.10. The Galois correspondence, as usual, can be used to analyze how one may build the extension, step-by-step, by adjoining solutions of differential equations of lower order, corresponding to taking intermediate extensions of the base field.…”

Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

“…Definition 2. 34 We say that a linear differential system ∂ (X) =ÃX, withà ∈ K n×n , is K-equivalent (or gauge equivalent over K) to a linear differential system ∂ (X) = AX, with A ∈ K n×n , if there exists B ∈ GL n (K) such that…”