Laurent polynomial identities on symmetric units of group algebras

Abstract

Let F be an infinite field of characteristic p ≠ 2 , G be a group, and * be an involution of G extended linearly to an involution of the group algebra FG. In the literature, group identities on units U ( FG ) and on symmetric units U + ( FG ) = { α ∈ U ( FG ) | α * = α } have been considered. Here, we investigate normalized Laurent polynomial identities (as a generalization of group identities) on U + ( FG ) under the conditions that either p > 2 or F is algebraically closed. For instance, we show that if G is torsion and U + ( FG ) satisfies a normalized Laurent polynomial identity, then U + ( FG ) satisfies a group identity and FG satisfies a polynomial identity.