Group rings whose symmetric elements are Lie nilpotent

Abstract

Let F G FG be the group ring of a group G G over a field F F , with characteristic different from 2 2 . Let ∗ \ast denote the natural involution on F G FG sending each group element to its inverse. Denote by ( F G ) + (FG)^{+} the set of symmetric elements with respect to this involution. A paper of Giambruno and Sehgal showed that provided G G has no 2 2 -elements, if ( F G ) + (FG)^{+} is Lie nilpotent, then so is F G FG . In this paper, we determine when ( F G ) + (FG)^{+} is Lie nilpotent, if G G does contain 2 2 -elements.