We consider a universe of finite Morley rank and the following definable objects: a field math formula, a non-trivial action of a group math formula on a connected abelian group V, and a torus T of G such that math formula. We prove that every T-minimal subgroup of V has Morley rank math formula. Moreover V is a direct sum of math formula-minimal subgroups of the form math formula, where W is T-minimal and ζ is an element of G of (...) order 4 inverting T. (shrink)

We study definable irreducible actions of SL₂(K) on an abelian group of Morley rank ≤ 3rk(K) and prove they are rational representations of the group.