Lie Algebras of Pro-Affine Algebraic Groups

Nahlus, Nazih

Canadian journal of mathematics, Tome 54 (2002) no. 3, pp. 595-607

Résumé

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 $\mathcal{L}(G)$ .

Détail

DOI : 10.4153/CJM-2002-021-9

Mots-clés : 14L, 16W, 17B45

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