Abstract view
Lie Algebras of ProAffine Algebraic Groups


Published:20020601
Printed: Jun 2002
Abstract
We extend the basic theory of Lie algebras of affine algebraic groups
to the case of proaffine algebraic groups over an algebraically
closed field $K$ of characteristic 0. However, some modifications
are needed in some extensions. So we introduce the prodiscrete
topology on the Lie algebra $\mathcal{L}(G)$ of the proaffine
algebraic group $G$ over $K$, which is discrete in the
finitedimensional case and linearly compact in general. As an
example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show
that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in
$\mathcal{L}(G)$.
We also discuss the Hopf algebra of representative functions $H(L)$ of
a 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 dense
in $\mathcal{L}(G)$.