Extensions of differential representations of SL(2) and tori

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. The 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.Comment: 21 pages; few misprints corrected; Lemma 4.6 adde

Zariski Closures of Reductive Linear Differential Algebraic Groups

Linear differential algebraic groups (LDAGs) appear as Galois groups of systems of linear differential and difference equations with parameters. These groups measure differential-algebraic dependencies among solutions of the equations. LDAGs are now also used in factoring partial differential operators. In this paper, we study Zariski closures of LDAGs. In particular, we give a Tannakian characterization of algebraic groups that are Zariski closures of a given LDAG. Moreover, we show that the Zariski closures that correspond to representations of minimal dimension of a reductive LDAG are all isomorphic. In addition, we give a Tannakian description of simple LDAGs. This substantially extends the classical results of P. Cassidy and, we hope, will have an impact on developing algorithms that compute differential Galois groups of the above equations and factoring partial differential operators.Comment: 26 pages, more detailed proof of Proposition 4.

Reductive inear differential algebraic groups and the Galois groups of parameterized linear differential equations

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending existing results, which were obtained for SL(2) in the case of just one derivation. As an application of the above bound, we develop an algorithm that tests whether the parameterized differential Galois group of a system of linear differential equations is reductive and, if it is, calculates it.Comment: 61 page