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Lie Algebras of Pro-Affine Algebraic Groups

Open Access article
 Printed: Jun 2002
  • Nazih Nahlus
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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 $[L,L]$ 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, 16W, 17B45 show english descriptions unknown classification 14L
unknown classification 16W
Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]
14L - unknown classification 14L
16W - unknown classification 16W
17B45 - Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]

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