On the structure of a real crossed group algebra

Abstract

The main result of this paper is that there exist non-principal left ideals in a certain twisted group algebraAof the infinite dihedral group ≤a, b|b−1ab=a−1,b2= 1 ≥ over the field R of real numbers: namely in theAdefined byb−lab = a−1, b2= −1, and λa=aλ, λb=bλ for all real λ.The motivation comes from the study (in a series of papers by Berman and the author) of finitely generated torsion-free RG-modules for groupsGwhich have an infinite cyclic subgroup of finite index. In a sense, this amounts to studying modules over (full matrix algebras over) a finite set of R-algebras [namely, for the groups in question, these algebras take on the role played by R, C and H (the real quaternions) in the theory of real representations of finite groups]. For all but two algebras in that finite set, satisfying results have been obtained by exploiting the fact that each of them is either a ring with zero divisors or a principal left ideal ring. The other two are known to have no zero divisors. One of them is the presentA. The point of the main result is that new ideas will be needed for understandingA-modules.A number of subsidiary results are concerned with convenient generating sets for left ideals inA.