Division Algebras Generated by Finitely Generated Nilpotent Groups
نویسنده
چکیده
Division 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/A, where A is the FC-center of G, (ii) the division subalgebra K(A) of D generated by A, and (iii) the subgroup K(d)*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.
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