Products of unipotent elements in certain algebras

Mai Hoang Bien; Peter V. Danchev; Mojtaba Ramezan-Nassab; Tran Nam Son

doi:10.1515/forum-2023-0005

Article

Products of unipotent elements in certain algebras

Mai Hoang Bien , Peter V. Danchev , Mojtaba Ramezan-Nassab and Tran Nam Son

Published/Copyright: May 13, 2023

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From the journal Forum Mathematicum Volume 35 Issue 6

Abstract

Let F be an algebraically closed field and let R be a locally finite algebra over F. This paper aims to show that any element of R is a product of at most three unipotent elements from R if and only if the element lies in the first derived subgroup of the unit group of R. In addition, this necessary and sufficient condition is applied to twisted group algebras of locally finite groups over a field of either zero characteristic or characteristic p ≠ 0 for some prime p. Moreover, we explore some crucial properties satisfied by certain algebras like the conditions concerned with the connections between unipotent elements of index 2 and commutators, as well as we investigate the unipotent radical of some subgroups of a finite-dimensional algebra R over a field F with at least four elements. In particular, we again apply these results to twisted group algebras.

Keywords: Locally finite(-dimensional) algebras; group algebras; commutators; unipotents; fields; division rings

MSC 2020: 15B33; 16U60; 16K40; 20C07

Communicated by Manfred Droste

Funding source: University of Science, Vietnam National University, Hanoi

Award Identifier / Grant number: T2023-01

Funding source: Bulgarian National Science Fund

Award Identifier / Grant number: KP-06 No. 32/1

Funding source: Junta de Andalucía

Award Identifier / Grant number: FQM 264

Funding source: Türkiye Bilimsel ve Teknolojik Araştirma Kurumu

Award Identifier / Grant number: BIDEB 2221

Funding source: Institute for Research in Fundamental Sciences

Award Identifier / Grant number: 1402160023

Funding source: Vingroup Innovation Foundation

Award Identifier / Grant number: VINIF.2022.TS105

Funding statement: The scientific work of the first-named author (M. H. Bien) is funded by University of Science, VNU-HCM (Grant No. T2023-01). The scientific work of the second-named author (P. V. Danchev) is partially supported by the Bulgarian National Science Fund (Grant No. KP-06 No. 32/1 of December 07, 2019) as well as by the Junta de Andalucía (Grant No. FQM 264) and by the BIDEB 2221 of TÜBİTAK. The research work of the third-named author (M. Ramezan-Nassab) was in part supported by a grant from IPM (Grant No. 1402160023). Likewise, the fourth-named author (T. N. Son) was funded by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF, Grant No. VINIF.2022.TS105).

Acknowledgements

The authors are very grateful to the specialist referee for the given numerous expert comments and suggestions that helped to improve the final version of the current article.

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Received: 2023-01-09

Revised: 2023-04-18

Published Online: 2023-05-13

Published in Print: 2023-11-01

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Articles in the same Issue

https://doi.org/10.1515/forum-2023-0005

Keywords for this article

Locally finite(-dimensional) algebras; group algebras; commutators; unipotents; fields; division rings