Free algebras in division rings with an involution
Vitor O. Ferreira, \'Erica Z. Fornaroli, Jairo Z. Gon\c{c}alves
TL;DR
This paper provides criteria for constructing explicit free algebras within division rings with involution, applying these to specific cases like group algebras and Weyl algebras to identify free subalgebras generated by symmetric elements.
Contribution
It introduces general criteria for finding free algebras inside division rings with involution and demonstrates their application to notable algebraic structures.
Findings
Existence of free subalgebras generated by symmetric elements in specific division rings.
Criteria applicable to division rings of group algebras and Weyl algebras.
Explicit construction of free algebras in these contexts.
Abstract
Some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented. These criteria are applied to some special cases of division rings with natural involutions, yielding, for instance, free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first Weyl algebra.
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