Products of unipotent elements in certain algebras

Abstract

AbstractLetFbe an algebraically closed field and letRbe a locally finite algebra overF. This paper aims to show that any element ofRis a product of at most three unipotent elements fromRif and only if the element lies in the first derived subgroup of the unit group ofR. 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 characteristicp≠0{p\not=0}for some primep. 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 algebraRover a fieldFwith at least four elements. In particular, we again apply these results to twisted group algebras.