Algebraische Gruppen by Jan Draisma

By Jan Draisma

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Dies gibt eine andere Interpretation der Lie-Klammer in L(G). Kurz geht das wie folgt: f¨ ur g ∈ G schreiben wir cg : G → G f¨ ur die Konjugation h → ghg −1 mit g. Die bildet e auf sich selbst ab, und deshalb ist das Differential de cg eine lineare Abbildung Te G → Te G, die wir mit Ad(g) bezeichnen. Dies gibt also eine Abbildung Ad : G → End(Te G). 1. Sei G = GLn und identifiziere Te G mit Mn . Dann ist cg die Einschr¨ ankung auf die offene dichte Teilmenge GLn ⊆ Mn der linearen Abbildung Mn → Mn , A → gAg −1 .

Dann schreiben wir Lie(G) f¨ ur die Menge aller Derivationen D ∈ Der(O(G)) wof¨ ur gilt Dλg = λg D f¨ ur alle g ∈ G, wo λg ∈ Aut(O(G)) die Linksverschiebung mit g ist. ]. Die Elemente von Lie(G) heissen links-invariante Derivationen. 37 38 7. 48. (1) Sei G = Ga die additive Gruppe, mit O(G) = K[t]. ∂ ∂ Es gilt Der(O(G)) = K[t] ∂t . Sei D = f ∂t mit f ∈ K[t]. Dann gilt f¨ ur s ∈ K einerseits (λs Dt)(0) = (λs f )(0) = f (−s) und andererseits (Dλs t)(0) = (D(t − s))(0) = (f − 0)(0) = f (0); D ist also genau dann links-invariant, wenn f konstant ist.

Ist char K = p > 0, so hat L(G) noch eine extra Struktur: Man u uft ¨berpr¨ leicht, dass in diesem Fall die p-te Potenz einer Derivation wieder eine Derivation ist, und dass Lie(G) abgeschlossen ist under p-te Potenzen Nehmen. Dies induziert eine entsprechende Struktur auf L(G); wir werden uns aber nicht allzu sehr damit besch¨ aftigen. Nun da wir zu einer algebraischen Gruppe G eine Lie-Algebra L(G) konstruiert haben, m¨ ochten wir einsehen, dass diese Konstruktion vertr¨aglich ist mit Homomorphismen algebraischer Gruppen.

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