By W. W. Sawyer
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4, Sec. 8]). If N is nilpotent, then exp(N ) is a finite sum of at most n terms, and so the map exp : N n → GLn is a morphism where N n ⊆ Mn denotes the closed subset of nilpotent matrices. A matrix A is called unipotent if A − E is nilpotent, or equivalently, if all eigenvalues ∞ 1 k k=0 k! 2. HOMOMORPHISMS AND EXPONENTIAL MAP 31 are equal to 1. Thus the set Un = E + N n ⊆ GLn of unipotent matrices is a closed subset, and the image of N n under exp is contained in Un . n−1 1 k k=0 k! 1. Proposition.
Exercise. (1) Show that every morphism ϕ : C∗ → C∗ such that ϕ(1) = 1 is a group homomorphism. Determine the automorphism group of C∗ (as an algebraic group). (2) Show that every nontrivial group homomorphism C+ → C+ is an isomorphism and determine the automorphism group of C+ . prop The proposition above has the following consequence which is the well-known mapping property in standard group theory. 10. Proposition. Let G, G , H be algebraic groups, ϕ : G → G a surjective homomorphism and µ : G → H a homomorphism such that ker µ ⊇ ker ϕ.
Exercise. Define the polynomials n En (x) := k=0 1 k x k! n and Ln (x) := k=1 (−1)k−1 (x − 1)k k n+1 and show that E(L(x)) = x mod x and L(E(x)) = x mod xn+1 . z (Hint: For all z ∈ C we have e = Ln (z) + z n+1 h(z) with a holomorphic function h, and for all y in a neighborhood U of 1 ∈ C we have ln(y) = Ln (y) + (y − 1)n+1 g(y) with g holomorphic in U . prop Exercise. Let U be a unipotent group. Then the power maps pm : U → U for m = 0 are isomorphisms of varieties . (Hint: This is clear for U C+ .