= 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.
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