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Lie Algebras of Pro-Affine Algebraic Groups

Published online by Cambridge University Press:  20 November 2018

Nazih Nahlus
Nazih Nahlus*
Affiliation:
Mathematics Department, American University of Beirut, Beirut, Lebanon, email: nahlus@aub.edu.lb
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Abstract

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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 $\mathcal{L}(G)$.

Keywords

14L16W17B45
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 3 , 01 June 2002 , pp. 595 - 607
Copyright
Copyright © Canadian Mathematical Society 2002

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