Group-graded Ore extensions and graded simplicity

Abstract

We study Ore extensions $R[t;\sigma ,\delta ]$ of rings graded by an arbitrary group $\Gamma$. The grading of $R$ extends to a grading with $t$ homogeneous of degree $\gamma$ if and only if $\sigma$ and $\delta$ are compatible with conjugation and translation by $\gamma$, and this grading is then unique. For nonabelian $\Gamma$ the degree of $t$ need not centralize the support of $R$; this twist can be removed by a change of variable when $R_{\gamma }$ contains a unit, but not in general. The graded forms of the classical necessary conditions for simplicity are not sufficient, as the quantized Weyl algebra shows. If $R$ is graded-simple and $\sigma$ is an automorphism, then $R[t;\sigma ,\delta ]$ is graded-simple if and only if it contains no homogeneous normal element that is monic of positive $t$-degree. For Ore extensions of group algebras, an affine action of the group on the line describes all graded ideals; in characteristic zero, graded simplicity is equivalent to the nonvanishing of a class in $H^1(G,k_{\chi })$, and simplicity forces the group to be metabelian.