Idempotents in the group algebra of the infinite dihedral group

Abstract

We prove that over an algebraically closed field $\mathbb{K}$ of characteristic different from $2$, the group algebra $R=\mathbb{K}{D}_{\infty }$ of the infinite dihedral group $D_{\infty }$ has exactly six conjugacy classes of involutions (equivalently, of idempotents). This allows us to recover the fact that $R$ admits exactly four non-isomorphic indecomposable projective modules of the form $eR$ where $e$ is an idempotent, a result that was first established by Berman and Buzási.

6 pages