Lie Algebras of Pro-Affine Algebraic Groups

摘要

We extend the basic theory of Lie algebras of affine algebraic groupsto the case of pro-affine algebraic groups over an algebraicallyclosed field $K$ of characteristic 0. However, some modificationsare needed in some extensions. So we introduce the pro-discretetopology on the Lie algebra $mathcal{L}(G)$ of the pro-affinealgebraic group $G$ over $K$, which is discrete in thefinite-dimensional case and linearly compact in general. As an example, if $L$ is any sub Lie algebra of $mathcal{L}(G)$, we showthat the closure of $[L,L]$ in $mathcal{L}(G)$ is algebraic in$mathcal{L}(G)$. We also discuss the Hopf algebra of representative functions $H(L)$ ofa residually finite dimensional Lie algebra $L$. As an example, we show that if $L$ is a sub Lie algebra of $mathcal{L}(G)$ and $G$is connected, then the canonical Hopf algebra morphism from $K[G]$into $H(L)$ is injective if and only if $L$ is algebraically densein $mathcal{L}(G)$.