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A Hopf algebra is a compatible algebra and coalgebra equipped with a counit and antipode. Coordinate rings of affine algebraic groups are commutative Hopf algebras.

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.

The center consists of scalar symplectic transformations:This description retains the nonreduced center in characteristic two. Since $Z_G$ is a finite central subgroup scheme, the invariant ring $k[G{]}^{Z_G}$ is finitely generated andis an affine algebraic group; the quotient map is finite and faithfully flat.

The quotient torus is $T/{\mu }_2$. Its character and cocharacter lattices areThe roots and coroots are the same type-$C_n$ sets written in part (ii), now regarded in these lattices. This is the adjoint root datum of type $C_n$.

Solved by gpt-5.6-sol high.