Invariant ideals of Abelian group algebras under the torus action of a field. I.
Let \(K\) be a field, \(V\) an Abelian group, \(G\) a group of automorphisms of \(V\). 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 the papers [\textit{C. J. B. Brookes, D. M. Evans}, Math. Proc. Camb. Philos. Soc. 130, No. 2, 287-294 (2001; Zbl 1005.20005), \textit{D. S. Passman, A. E. Zalesskiĭ}, Proc. Am. Math. Soc. 130, No. 4, 939-949 (2002; Zbl 0992.16021) and \textit{J. M. Osterburg, D. S. Passman, A. E. Zalesskiĭ}, Proc. Am. Math. Soc. 130, No. 4, 951-957 (2002; Zbl 0992.16022)] the special case, when \(V\) is a vector space over an infinite field and \(G\) is the multiplicative group of scalars, was settled. Let \(V=V_1\oplus V_2\) be a finite-dimensional vector space over an infinite locally-finite field \(F\). Then \(V\) admits the torus action of \(G=F^\bullet\) by defining \((v_1\oplus v_2)^g=v_1g^{-1}\oplus v_2g\). If \(K\) is a field of characteristic different from that of \(F\), then \(G\) acts on the group algebra \(K[V]\). In this paper, the case when \(V_1\) and \(V_2\) are both 1-dimensional is examined and the author shows that there are just four \(G\)-stable proper ideals of \(K[V]\), namely, with \(A\) a subgroup of \(V\), the augmentation ideal \(\omega(A,V)\) of \(A\), the kernel of the natural epimorphism \(K[V]\to K[V/A]\), these are \(\omega(V,V)\), \(\omega(V_1,V)\), \(\omega(V_2,V)\) and \(\omega(V_1,V)\cap\omega(V_2,V)\). Furthermore, it is announced that, applying the same techniques, the analogous problem for the torus action of the special linear group on the coordinate space of dimension \(n\) can be handled analogously, which will be the subject of a forthcoming paper.
scientific article; zbMATH DE number 5853359
scientific article