Exercise 18, § 6

Let $A$ be a finite-dimensional connected commutative complex Lie group with Lie algebra $a$; let $\Lambda$ be the kernel of $\exp_A$, so that $\Lambda$ is identified with $a/\Lambda$.

(a) The following conditions are equivalent: (a1) The canonical mapping $\mathbf{C} \otimes_{\mathbf{Z}} \Lambda \to a$ is injective. (a2) $A$ is isomorphic to a Lie subgroup of some $(\mathbf{C}^*)^n$. (a3) $A$ is isomorphic to a group $(\mathbf{C}^*)^p \times \mathbf{C}^q$. (a4) $A$ has a finite-dimensional faithful complex linear representation. (a5) $A$ has a finite-dimensional semi-simple faithful complex linear representation with closed image. (b) The following conditions are equivalent: (b1) The canonical mapping $\mathbf{C} \otimes_{\mathbf{Z}} \Lambda \to a$ is surjective. (b2) $A$ is isomorphic to a quotient of a group $(\mathbf{C}^*)^n$. (b3) No direct factor of $A$ is isomorphic to $\mathbf{C}$. (b4) Every complex linear representation of $A$ is semi-simple. (c) The following conditions are equivalent: (c1) The canonical mapping $\mathbf{C} \otimes_{\mathbf{Z}} \Lambda \to a$ is bijective. (c2) $A$ is isomorphic to a group $(\mathbf{C}^*)^n$. (d) Let $F$ be a finite subgroup of $A$ and let $A' = A/F$. Show that $A$ satisfies conditions (a_i) (resp. (b_i), (c_i)) if and only if $A'$ does.