Unitary units and skew elements in group algebras.
Let \(FG\) be the group algebra of a group \(G\) over a field \(F\) and let * denote the canonical involution of \(FG\) induced by the map \(g\mapsto g^{-1}\), \(g\in G\). The unitary units in \(FG\), i.e., the elements \(u\in FG\) for which \(uu^*=1\), form a subgroup \(\text{Un}(FG)\) of the group of units of \(FG\). This paper studies the property of \(\text{Un}(FG)\) being such that it satisfies a group identity. A complete classification is given for the case when \(\text{char\,}F=0\), \(G\) is a torsion group and the group identity is such that it does not become trivial when evaluated at elements of order \(2\). Let \(FG^-\) be the Lie algebra consisting of the skew elements, i.e., the elements \(x\in FG\) for which \(x^*=-x\). The main result on skew elements is a characterization for \(FG^-\) to be Lie nilpotent, provided that \(\text{char\,}F\neq 2\) and \(G\) is such that \(FG\) is semiprime.
scientific article; zbMATH DE number 1965483