Groups and noncommutative algebra: interactions and applications
Free symmetric and unitary pairs in division rings with involution.
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| Author(s): |
Total Authors: 3
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| Affiliation: | [1] Univ Sao Paulo, Dept Math, BR-05508090 Sao Paulo, SP - Brazil
Total Affiliations: 1
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| Document type: | Journal article |
| Source: | ALGEBRAS AND REPRESENTATION THEORY; v. 23, n. 3, p. 605-619, JUN 2020. |
| Web of Science Citations: | 0 |
| Abstract | |
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) | |
| FAPESP's process: | 15/09162-9 - Non commutative algebra and applications |
| Grantee: | Francisco Cesar Polcino Milies |
| Support Opportunities: | Research Projects - Thematic Grants |