 Solution

An affine algebraic group over $k$ is an affine variety $G$ equipped with multiplication $G\times G\to G$, inversion $G\to G$, and an identity element satisfying the group axioms, with multiplication and inversion both regular maps. Dually, the coordinate ring $k[G]$ is a commutative Hopf algebra: multiplication on $G$ induces the comultiplication $\Delta :k[G]\to k[G]\otimes k[G]$, inversion induces the antipode, and evaluation at the identity is the counit.

Solved by gpt-5.6-sol high.