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Moreover, for h ∈ H we have ωΦ(h) = ϕ(ω)Φ(h) = ϕ(ωh) = ϕ(ω) = ω, hence H is Φ-invariant, and thus Φ|H is a Frobenius endomorphism. For ω ∈ Ωϕ arbitrary let now g ∈ G such that ωg = ω . Then we have ωg = ω = ϕ(ω ) = ϕ(ωg) = ϕ(ω)Φ(g) = ωΦ(g), hence gΦ(g −1 ) ∈ H. Thus there is h ∈ H such that gΦ(g −1 ) = h−1 Φ(h), implying hg = Φ(hg) ∈ GΦ and ω = ωhg. 40). 41). 1) Proposition. Let G be an algebraic group, let Vλ irreducible affine varieties together with morphisms ϕλ : Vλ → G such that 1G ∈ Wλ := ϕλ (Vλ ), for all λ ∈ Λ, where Λ is an index set.

29). 3) Example: The additive and the multiplicative group. a) Let G := Ga = K be the additive group, hence K[Ga ] ∼ = K[X]. Thus we have dimK (T0 (Ga )) = 1, hence T0 (Ga ) is a commutative Lie algebra. From ∂λ∗−x (X) = ∂(X + x) = 1 = λ∗−x ∂(X), for all x ∈ G, we deduce that ∂ is left invariant, hence L(Ga ) = ∂ K . b) Let G := Gm = KX be the multiplicative group, hence K[Gm ] ∼ = K[X]X . Thus we have dimK (T1 (Gm )) = 1, hence T1 (Gm ) is a commutative Lie algebra. We have L(Gm ) ∼ = T1 (Gm ) ∼ = DerK (K[X]X , K1 ) ∼ = DerK (K[X], K1 ) ∼ = DerK (K[X] X−1 , K1 ) ∼ = HomK ( X − 1 / X − 1 2 , K1 ).

S−1 , λs + 1, λs+1 , . . , λn ] max λ, where 1 ≤ r < s ≤ n. Hence there are representatives diag[Jλr (1), Jλs (1), B] ∈ Cλ and II Algebraic groups 34 diag[Jλr −1 (1), Jλs +1 (1), B] ∈ Cµ , where m := λr + λs ≤ n and B ∈ Un−m . Let µ := [λr − 1, λs + 1] m and λ := [λr , λs ] m. Hence we have µ λ as well as diag[Jλr −1 (1), Jλs +1 (1)] ∈ Cµ ⊆ (SLm )u and diag[Jλr (1), Jλs (1)] ∈ Cλ ⊆ (SLm )u . It is immediate that SLm → G : A → diag[A, Em−n ] is a closed embedding of algebraic groups, which extends to a closed embedding of affine varieties SLm × {B} → G : [A, B] → diag[A, B], where B ∈ Un−m is as above.

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