Let $k$ be a field, $K$ a finite extension of $k$, and $K_{n}^{M}(K)$ the $n$-th Milnor K-group of $K$, that is, $$ K_{n}^{M}(K)=K^{\times}\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} K^{\times}/I, $$ where $I$ is the subgroup generated by $\{a_{1}\otimes\cdots\otimes a_{n}\mathrel{\vert}a_{i}+a_{j}=1 \text{ for some }i\neq j \}$.

Question: Are there a Hopf algebra $A$ over $k$ such that ${\rm Hom}_{k}(A,K)=K_{n}^{M}(K)$  ? In other words, are Milnor K-groups affine algebraic groups  ?

Of course when $n=1$, we can take $A=k[t^{\pm}]$. So I am interested in the case of $n\geq 2$. Do you have any positive or negative answers about this?

Let $k$ be a field, $K$ a finite extension of $k$, and $K_{n}^{M}(K)$ the $n$-th Milnor K-group of $K$, that is, $$ K_{n}^{M}(K)=K^{\times}\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} K^{\times}/I, $$ where $I$ is the subgroup generated by $\{a_{1}\otimes\cdots\otimes a_{n}\mathrel{\vert}a_{i}+a_{j}=1 \text{ for some }i\neq j \}$.

Question: Is there a Hopf algebra $A$ over $k$ such that ${\rm Hom}_{k}(A,K)=K_{n}^{M}(K)$? In other words, are Milnor K-groups affine algebraic groups?

Of course when $n=1$, we can take $A=k[t^{\pm}]$. So I am interested in the case of $n\geq 2$. Do you have any positive or negative answers about this?

Let $k$ be a field and $K$ a finite extension of $k$. $K_{n}^{M}(K)$ is $n$-th Milnor K-group of $K$, this is, $$ K_{n}^{M}(K)=K^{\times}\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} K^{\times}/I, $$ where $I$ is subgroup generated by $\{a_{1}\otimes\cdots\otimes a_{n}\mathrel{\vert}a_{i}+a_{j}=1 \text{ for some }i\neq j \}$.

Question. Are there a Hopf algebra $A$ over $k$ such that ${\rm Hom}_{k}(A,K)=K_{n}^{M}(K)$ ? in otherword Are Milnor K-groups affine algebraic groups ?

ofcourse when $n=1$, we can take $A=k[t^{\pm}]$. So I have interest in the cases of $n\geq 2$. Have you any positive or negative answers about this  ?

Let $k$ be a field, $K$ a finite extension of $k$, and $K_{n}^{M}(K)$ the $n$-th Milnor K-group of $K$, that is, $$ K_{n}^{M}(K)=K^{\times}\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} K^{\times}/I, $$ where $I$ is the subgroup generated by $\{a_{1}\otimes\cdots\otimes a_{n}\mathrel{\vert}a_{i}+a_{j}=1 \text{ for some }i\neq j \}$.

Question: Are there a Hopf algebra $A$ over $k$ such that ${\rm Hom}_{k}(A,K)=K_{n}^{M}(K)$ ? In other words, are Milnor K-groups affine algebraic groups ?

Of course when $n=1$, we can take $A=k[t^{\pm}]$. So I am interested in the case of $n\geq 2$. Do you have any positive or negative answers about this?