Canadian Mathematical Society www.cms.math.ca
Abstract view

# Lie Algebras of Pro-Affine Algebraic Groups

Published:2002-06-01
Printed: Jun 2002
• Nazih Nahlus
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
 Format: HTML LaTeX MathJax PDF PostScript

## Abstract

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 $[L,L]$ 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 $\mathcal{L}(G)$.
 MSC Classifications: 14L - unknown classification 14L16W - unknown classification 16W17B45 - Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]

© Canadian Mathematical Society, 2013 : http://www.cms.math.ca/