arXiv subjects
Publications and source records attributed to Ramy Yammine.
Let $H$ be a connected Hopf algebra acting on an algebra $A$. Working over a base field having characteristic $0$, we show that for a given prime (semi-prime, completely prime) ideal $I$ of $A$, the largest $H$-stable ideal of A contained in $I$ is also prime (semi-prime, completely prime). We also prove a similar result for certain subrings of convolution algebras.
Let $H$ be a cocommutative Hopf algebra acting on an algebra $A$. Assuming the base field to be algebraically closed and the $H$-action on $A$ to be integral, that is, it is given by a coaction of some Hopf subalgebra of the finite dual $H^\circ$ that is an integral domain, we stratify the prime spectrum $\mbox{Spec}\, A$ in terms of the prime spectra of certain commutative algebras. For arbitrary $H$-actions in characteristic $0$, we show that the largest $H$-stable ideal of $A$ that is contained in a given semiprime ideal of $A$ is semiprime as well.
We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{\text{adfin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely generated as module over a cocommutative Hopf subalgebra), we show that $H_{\text{adfin}}$ is a Hopf subalgebra of $H$. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, $H_{\text{adfin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.