Commutative Hopf algebras over a field

Let $k$ be a field (Field). A commutative Hopf algebra over $k$ is a commutative unital $k$-algebra $A$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms, Commutative ring) together with $k$-algebra homomorphisms (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$) $\mathrm{\Delta }\text{\hspace{0.17em}⁣}:A\to A{\otimes }_{k}A,\epsilon \text{\hspace{0.17em}⁣}:A\to k,S\text{\hspace{0.17em}⁣}:A\to A,$ called the comultiplication, the counit and the antipode, such that, with $m_A$ the multiplication of $A$, with $u_A\text{\hspace{0.17em}⁣}:k\to A$ the unit of $A$, and with the canonical identifications $k{\otimes }_{k}A\cong A\cong A{\otimes }_{k}k$, the following identities hold.

1. $({\mathrm{id}}_A\otimes \mathrm{\Delta })\mathrm{\Delta }=(\mathrm{\Delta }\otimes {\mathrm{id}}_A)\mathrm{\Delta }$ (coassociativity).
2. $(\epsilon \otimes {\mathrm{id}}_A)\mathrm{\Delta }={\mathrm{id}}_A=({\mathrm{id}}_A\otimes \epsilon )\mathrm{\Delta }$ (counit identities).
3. $m_A(S\otimes {\mathrm{id}}_A)\mathrm{\Delta }=u_A\epsilon =m_A({\mathrm{id}}_A\otimes S)\mathrm{\Delta }$ (antipode identities).

Here $A{\otimes }_{k}A$ is the tensor product of $A$ with itself over $k$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums), whose elements are finite sums ${\sum }_i{a}_i\otimes b_i$; it carries the $k$-algebra structure making it the coproduct of $A$ with itself among commutative $k$-algebras, so that a $k$-algebra homomorphism out of $A{\otimes }_{k}A$ is exactly a pair of $k$-algebra homomorphisms out of $A$ (Universal mapping property of the tensor product of commutative algebras).

A morphism of commutative Hopf algebras $f\text{\hspace{0.17em}⁣}:(A,{\mathrm{\Delta }}_A,{\epsilon }_A,S_A)\to (B,{\mathrm{\Delta }}_B,{\epsilon }_B,S_B)$ is a $k$-algebra homomorphism with $(f\otimes f){\mathrm{\Delta }}_A={\mathrm{\Delta }}_{B}f$, ${\epsilon }_{B}f={\epsilon }_A$ and $f{S}_A=S_{B}f$; morphisms are required to preserve all three structure maps, not only the comultiplication.

The Hopf algebra is finitely generated if $A$ is a finitely generated $k$-algebra (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras). Neither reducedness, nor smoothness, nor finite generation is imposed by the definition, and $k$ is an arbitrary field.