Prime ideals in enveloping rings

Abstract:

Let $L$ be a Lie algebra over the field $K$ of characteristic $0$ and let $U(L)$ denote its universal enveloping algebra. If $R$ is a $K$-algebra and $L$ acts on $R$ as derivations, then there is a natural ring generated by $R$ and $U(L)$ which is denoted by $R\# U(L)$ and called the smash product of $R$ by $U(L)$. The aim of this paper is to describe the prime ideals of this algebra when it is Noetherian. Specifically we show that there exists a twisted enveloping algebra $U(X)$ on which $L$ acts and a precisely defined one-to-one correspondence between the primes $P$ of $R\#U(L)$ with $P \cap R = 0$ and the $L$-stable primes of $U(X)$. Here $X$ is a Lie algebra over some field $C \supseteq K$.