01 February 2017; 12:00 room c3b/158
Javier Sánchez. A way of obtaining free group algebras inside division rings

Abstract: In the mid eighties, L. Makar-Limanov conjectured the following: Let D be a division ring with center Z. If D is finitely generated (as a division ring) over Z and [D:Z]=\infty, then D contains a noncommutative free Z-algebra. In many of the examples for which the conjecture is known to be valid, the division ring D contains a (noncommutative) free group algebra over Z, not only a free Z-algebra. Note that, in general, if X is the set of free generators of a free algebra inside a division ring, X may not be a set of free generators of a free group Z-algebra. Given a division ring D with a valuation $upsilon$, we obtain sufficient conditions for the existence of noncommutative free group algebras in D. These conditions involve the graded division ring grad_{upsilon}(D) associated to the filtration induced by the valuation.