Division algebras generated by finitely generated nilpotent groups

AbstractDivision algebras D generated by some finitely generated nilpotent subgroup G of the multiplicative group D∗ of D are studied and the question to what extent G is determined by D is considered. Trivial examples show that D does not determine G up to isomorphism. However, it is proved that if F denotes the center of D, then the F-subalgebra of D generated by G is in fact determined up to isomorphism by D. Using the structure of this subalgebra it is further concluded that D does at least determine (i) the group G/Δ, where Δ is the FC-center of G, (ii) the division subalgebra K(Δ) of D generated by Δ, and (iii) the subgroup K(Δ)∗G of D∗. The principal technical tools are the so-called (crossed) Hilbert-Neumann rings of ordered groups over rings which are also studied here in their own right