Free Symmetric and Unitary Pairs in the Field of Fractions of Torsion-Free Nilpotent Group Algebras

Autor(es)

Ferreira, Vitor O. ^[1] ; Goncalves, Jairo Z. ^[1] ; Sanchez, Javier ^[1]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, Dept Math, BR-05508090 Sao Paulo, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: ALGEBRAS AND REPRESENTATION THEORY; v. 23, n. 3, p. 605-619, JUN 2020.
• Citações Web of Science: 0

Resumo

Let k be a field of characteristic different from 2 and let G be a nonabelian residually torsion-free nilpotent group. It is known that G is an orderable group. Let k(G) denote the subdivision ring of the Malcev-Neumann series ring generated by the group algebra of G over k. If {*} is an involution on G, then it extends to a unique k-involution on k(G). We show that k(G) contains pairs of symmetric elements with respect to {*} which generate a free group inside the multiplicative group of k(G). Free unitary pairs also exist if G is torsion-free nilpotent. Finally, we consider the general case of a division ring D, with a k-involution {*}, containing a normal subgroup N in its multiplicative group, such that G subset of N, with G a nilpotent-by-finite torsion-free subgroup that is not abelian-by-finite, satisfying G{*} = G and N{*} = N. We prove that N contains a free symmetric pair. (AU)

• Processo FAPESP: 15/09162-9 - Álgebra não comutativa e aplicações
• Beneficiário: Francisco Cesar Polcino Milies
• Modalidade de apoio: Auxílio à Pesquisa - Temático