Invariant ideals of abelian group algebras under the torus action of a field, I

Abstract

Let V = V 1 ⊕ V 2 be a finite-dimensional vector space over an infinite locally-finite field F. Then V admits the torus action of G = F • by defining ( v 1 ⊕ v 2 ) g = v 1 g − 1 ⊕ v 2 g . If K is a field of characteristic different from that of F, then G acts on the group algebra K [ V ] and it is an interesting problem to determine all G-stable ideals of this algebra. In this paper, we consider the special case when V 1 and V 2 are both 1-dimensional and we show that there are just four G-stable proper ideals of K [ V ] .