A Problem on Affine Algebraic Groups and Hopf Algebra Structures
Problem: (Definition) An affine algebraic group $G$ is an affine algebraic variety (in $\mathbb {A}^n_k$, for a given $n\in \mathbb {N}$) with group structure, such that the multiplication and the inversion, from $G$ to $G$, are algebraic variety morphisms.
i) Show that the symplectic group $Sp(2,k)$, given by $x\in Gl(2,k)$ such that $x^t\cdot \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \cdot x=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, is an affine algebraic group.
ii) Show that the ring of regular functions (or coordinate ring) $A:=k[G]$, seen here as a $k$-algebra and with $G$ being an affine algebraic group, satisfies the following: there exists $\mu: A\otimes A\rightarrow A, i:A\rightarrow A$ and $e$: such that the attached diagrams are commutative. In other words, show that $k[G]$ is a Hopf algebra with identity.
i) Show that the symplectic group $Sp(2,k)$, given by $x\in Gl(2,k)$ such that $x^t\cdot \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \cdot x=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, is an affine algebraic group.
ii) Show that the ring of regular functions (or coordinate ring) $A:=k[G]$, seen here as a $k$-algebra and with $G$ being an affine algebraic group, satisfies the following: there exists $\mu: A\otimes A\rightarrow A, i:A\rightarrow A$ and $e$: such that the attached diagrams are commutative. In other words, show that $k[G]$ is a Hopf algebra with identity.
Ering
93
The answer is accepted.
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