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Lie Algebras of Pro-Affine Algebraic Groups
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- Journal:
- Canadian Journal of Mathematics / Volume 54 / Issue 3 / 01 June 2002
- Published online by Cambridge University Press:
- 20 November 2018, pp. 595-607
- Print publication:
- 01 June 2002
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- Article
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We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field
$K$ of characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra 
$\mathcal{L}(G)$ of the pro-affine algebraic group 
$G$ over $K$, which is discrete in the finite-dimensional 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 
$\left[ L,\,L \right]$ 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 
$K\left[ G \right]$ into $H(L)$ is injective if and only if $L$ is algebraically dense in $\mathcal{L}(G)$.
