Characters Are Group-Like Elements
Let \(G = \operatorname{Spec} A\) be an affine group scheme over \(R\). The character group \(X^*(G) = \operatorname{Hom}_{\mathrm{grp}}(G, \mathbb{G}_m)\) (the definition, see Character and Cocharacter Lattices) admits the following computed description, in two steps:
- By the coordinate-ring bridge (Homomorphism of Affine Group Schemes), \[ \operatorname{Hom}_{\mathrm{grp}}(G, \mathbb{G}_m) \;\cong\; \operatorname{Hom}_{R\text{-bialg}}\bigl(R[\mathbb{Z}],\, A\bigr), \] since \(O(\mathbb{G}_m) = R[\mathbb{Z}]\) (The Diagonalizable Group Scheme D(M)).
- A bialgebra homomorphism \(\varphi : R[\mathbb{Z}] \to A\) is determined by \(u = \varphi(e^{1})\): multiplicativity forces \(\varphi(e^{n}) = u^n\), so \(u\) must be invertible, and compatibility with comultiplication and counit says exactly that \(u\) is group-like (\(\Delta u = u \otimes u\), \(\varepsilon(u) = 1\)). Conversely every group-like unit defines such a homomorphism. Hence \[ X^*(G) \;\cong\; \{\text{group-like units of } A\}, \] and when \(A\) is a Hopf algebra every group-like element is automatically a unit (its inverse is its antipode image), so \(X^*(G) \cong \mathrm{GroupLike}(A)\).
Under this identification the pointwise product of characters corresponds to the product of group-like elements (see Convolution Is Pointwise Multiplication).
Specialization to split tori. For \(A = R[M]\) over a domain \(R\), the group-like elements are exactly the basis elements \(e^m\), so \(X^*(D(M)) \cong M\) — this is the object half of Cartier duality recorded at Diagonalizable Groups Are Dual to Character Groups.
Proof
Step 1 is the bridge node. For step 2: given \(\varphi\), set \(u = \varphi(e^1)\); then \(\varphi(e^n) = u^n\) for \(n \geq 0\) by multiplicativity and for \(n < 0\) because \(e^{-1}\) is the inverse of \(e^1\), forcing \(u \in A^\times\). Compatibility with the group-like comultiplication of \(R[\mathbb{Z}]\) gives \(\Delta(u) = u \otimes u\) and \(\varepsilon(u) = 1\). Conversely a group-like unit \(u\) defines an algebra map \(R[\mathbb{Z}] \to A\), \(e^n \mapsto u^n\), which is a coalgebra map on the basis, hence a bialgebra map. The two constructions are mutually inverse. For a Hopf algebra, \(S(u)\,u = \varepsilon(u) = 1\) shows group-likes are units.