A Jordan–Hölder Theorem for differential algebraic groups

Abstract

We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a uniqueness result, prove several properties of almost simple groups and, in the ordinary differential case, classify almost simple linear differential algebraic groups.

Indecomposability for differential algebraic groups

“…So, X g := g G g −1 is n-indecomposable and contains the identity. By Theorem 3.5, the group generated by the family (X g ) g∈G is a strongly connected differential algebraic subgroup of G. P We should also note the result of Cassidy and Singer which says that if a strongly connected differential algebraic group is not commutative, then the differential type of the differential closure of derived subgroup is equal to the differential type of the whole group [6]. So, putting this together with the above theorem yields:…”

“…Generalizing this work to the difference-differential setting (or even more general settings) is of interest, but is not covered here. Also, though there are well-developed theories of numerical polynomials in these more general settings [14]; work along the lines of [6] (in those settings) would seem to be a prerequisite for proving results like those in this paper. The main original motivation for this work was the analysis of group theoretic properties of almost simple (and more generally strongly connected) differential algebraic groups.…”

“…For additional results on the properties of differential type and typical differential dimension, see [6] and [12].…”

“…Thus, this is a transitive action of H on a differential algebraic variety of differential type less than n. The centralizer of g in H must be a subgroup of H of differential type τ (H) and typical differential dimension equal to that of H. This is impossible, by the indecomposability of H, unless the kernel is simply H itself (see [6] or [10]). If that is the case, then by the transitivity of the action, |g H /N | = 1.…”

“…If that is the case, then by the transitivity of the action, |g H /N | = 1. P Cassidy and Singer make the following comment in [6], "We also do not have an example of a noncommutative almost simple linear differential algebraic group whose commutator subgroup is not closed in the Kolchin topology." The next result shows that such an example is not possible, even in the more general case of the group being strongly connected (with no assumption of linearity or almost simplicity).…”