Involutions and Anticommutativity in Group Rings

Abstract

AbstractLet g denote an involution on a group G. For any (commutative, associative) ring R (with 1), * extends linearly to an involution of the group ring RG. An element α ∊ RG is symmetric if α* = α and skew-symmetric if α* = -α. The skew-symmetric elements are closed under the Lie bracket, [αβ] = αβ - βα. In this paper, we investigate when this set is also closed under the ring product in RG. The symmetric elements are closed under the Jordan product, α˚α = αβ +βα. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.