Invariant ideals and polynomial forms

Abstract:

Let $K[\mathfrak H]$ denote the group algebra of an infinite locally finite group $\mathfrak H$. In recent years, the lattice of ideals of $K[\mathfrak H]$ has been extensively studied under the assumption that $\mathfrak H$ is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra $K[V]$ stable under the action of an appropriate automorphism group of $V$. Specifically, in this paper, we let ${\mathfrak {G}}$ be a quasi-simple group of Lie type defined over an infinite locally finite field $F$, and we let $V$ be a finite-dimensional vector space over a field $E$ of the same characteristic $p$. If ${\mathfrak {G}}$ acts nontrivially on $V$ by way of the homomorphism $\phi \colon {\mathfrak {G}}\to \mathrm {GL}(V)$, and if $V$ has no proper ${\mathfrak {G}}$-stable subgroups, then we show that the augmentation ideal $\omega K[V]$ is the unique proper ${\mathfrak {G}}$-stable ideal of $K[V]$ when ${\operatorname {char}} K\neq p$. The proof of this result requires, among other things, that we study characteristic $p$ division rings $D$, certain multiplicative subgroups $G$ of $D^{\bullet }$, and the action of $G$ on the group algebra $K[A]$, where $A$ is the additive group $D^{+}$. In particular, properties of the quasi-simple group ${\mathfrak {G}}$ come into play only in the final section of this paper.