Smash products and differential identities

Abstract:

Let $\mathbf {U}$ be the universal enveloping algebra of a Lie algebra and $R$ a $\mathbf {U}$-module algebra, where $\mathbf {U}$ is considered as a Hopf algebra canonically. We determine the centralizer of $R$ in $R\#\mathbf {U}$ with its associated graded algebra. We then apply this to the Ore extension $R[X;\phi ]$, where $\phi :X\to \mathrm {Der}(R)$. With the help of PBW-bases, the following is proved for a prime ring $R$: Let $Q$ be the symmetric Martindale quotient ring of $R$. For $f_i,g_i\in Q[X;\phi ]$, $\sum _if_irg_i=0$ for all $r\in R$ iff $\sum _if_i\otimes g_i=0$, where $\otimes$ is over the centralizer of $R$ in $Q[X;\phi ]$. Finally, we deduce from this Kharchenko’s theorem on differential identities.