Diagonalizable Groups Are Dual to Character Groups
The functor \[ D \longmapsto X(D)=\operatorname{Hom}(D,\mathbb{G}_m) \] gives an anti-equivalence between diagonalizable algebraic groups over an algebraically closed field and finitely generated abelian character groups of the allowed type in the given characteristic. Under this correspondence, groups with free abelian character group of finite rank are exactly the split tori; how the remaining invariants (smoothness, connectedness) read off the torsion of the character group is recorded where tori are defined.
For a finitely generated abelian group \(M\), the associated diagonalizable group is \[ D(M)=\operatorname{Spec} k[M], \] where \(k[M]\) is the group algebra with Hopf structure \(\Delta(e^m)=e^m\otimes e^m\).
Proof
For diagonalizable \(D\), the character decomposition of \(k[D]\) identifies the coordinate Hopf algebra with the group algebra \(k[X(D)]\). A morphism \(D_1\to D_2\) pulls characters back, giving a homomorphism \(X(D_2)\to X(D_1)\).
Conversely, any finitely generated abelian group \(M\) defines the Hopf algebra \(k[M]\), hence the diagonalizable group \(D(M)\). Its characters are exactly the basis elements indexed by \(M\). These two constructions are inverse to each other. Groups with torsion-free character group are precisely the split tori; note that in characteristic \(p > 0\) connectedness alone does not force torsion-freeness, since the character group \(\mathbb{Z}/p\) gives the connected non-smooth group \(\mu_p\).