Lie properties of symmetric elements in group rings.
Let \(KG\) be a group algebra and * its involution induced by an involution of the group. In the paper of the first and third authors [Commun. Algebra 21, No. 11, 4253-4261 (1993; Zbl 0797.16036)], it was proved that under suitable conditions if the set of symmetric elements (with respect to the classical involution) is Lie nilpotent then so is the group algebra. Recently research intensified concerning the question that which properties of symmetric or skew-symmetric elements infer the respective property of the whole algebra. The main results of the authors are as follows. If the characteristic is not 2 and there are no 2-elements in the group then the whole algebra is Lie \(n\)-Engel if and only if symmetric elements are Lie \(m\)-Engel; further, the whole algebra is Lie nilpotent if and only if symmetric elements are Lie-nilpotent. A key point in the arguments is the classical theorem of Amitsur that if the symmetric (or skew-symmetric) elements satisfy a polynomial identity then so does the ring. It is announced that in a paper by \textit{G.~T. Lee, E.~Spinelli} and \textit{S.~K.~Sehgal} [J. Pure Appl. Algebra 213, No. 6, 1173-1178 (2009; Zbl 1170.16019)] these results are established even in the presence of 2-elements in the group.
scientific article; zbMATH DE number 5549341
scientific article