A variation on Dixmier's counterexample concerning centralizers in $A_1$
I have asked this question in MO, but have not received any comments/answers, so now I ask it here:
This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ are commuting elements of the first Weyl algebra. Is it true that $P$ and $Q$ are polynomials in some third element $H$?"
It was answered in the negative (the counterexample seems to be due to Dixmier and Makar-Limanov).
Now replace the field of characteristic zero $k$ by a non-normal integral domain of characteristic zero $R$, for example $R=mathbb{C}[t^2,t^3]$.
Is it possible to adjust $P$ and $Q$ to some $tilde{P}$ and $tilde{Q}$ satisfying the following three conditions:
(i) $[tilde{P},tilde{Q}]=0$.
(ii) $tilde{P}$ and $tilde{Q}$ are not polynomials in some third element $H$.
(iii) $tilde{P}$ has a 'Dixmier mate', namely, there exists $B$ in the first Weyl algebra over $R$ such that $[tilde{P},B]=1$.
Remarks:
(1) The analogous question in $R[x,y]$ can be found here.
(2) In contrast, in $k[x,y]$, it is true that if $operatorname{Jac}(p,q)=0$, then $p=u(h)$ and $q=v(h)$, for some $h in k[x,y]$, $u(T),v(T) in k[T]$; see, for example, Corollary 1.3 in Nagata's paper ($k$ is an algebraically closed field of characteristic zero) or this question ($k$ is an arbitrary field).
$k$ can be replaced by any normal integral domain of characteristic zero, see this answer.
Any hints and comments are welcome!
ring-theory noncommutative-algebra
asked Nov 29 at 15:12
user237522
2,0871617
add a comment |
I have asked this question in MO, but have not received any comments/answers, so now I ask it here:
This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ are commuting elements of the first Weyl algebra. Is it true that $P$ and $Q$ are polynomials in some third element $H$?"
It was answered in the negative (the counterexample seems to be due to Dixmier and Makar-Limanov).
Now replace the field of characteristic zero $k$ by a non-normal integral domain of characteristic zero $R$, for example $R=mathbb{C}[t^2,t^3]$.
Is it possible to adjust $P$ and $Q$ to some $tilde{P}$ and $tilde{Q}$ satisfying the following three conditions:
(i) $[tilde{P},tilde{Q}]=0$.
(ii) $tilde{P}$ and $tilde{Q}$ are not polynomials in some third element $H$.
(iii) $tilde{P}$ has a 'Dixmier mate', namely, there exists $B$ in the first Weyl algebra over $R$ such that $[tilde{P},B]=1$.
Remarks:
(1) The analogous question in $R[x,y]$ can be found here.
(2) In contrast, in $k[x,y]$, it is true that if $operatorname{Jac}(p,q)=0$, then $p=u(h)$ and $q=v(h)$, for some $h in k[x,y]$, $u(T),v(T) in k[T]$; see, for example, Corollary 1.3 in Nagata's paper ($k$ is an algebraically closed field of characteristic zero) or this question ($k$ is an arbitrary field).
$k$ can be replaced by any normal integral domain of characteristic zero, see this answer.
Any hints and comments are welcome!
ring-theory noncommutative-algebra
asked Nov 29 at 15:12
user237522
2,0871617
add a comment |
I have asked this question in MO, but have not received any comments/answers, so now I ask it here:
This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ are commuting elements of the first Weyl algebra. Is it true that $P$ and $Q$ are polynomials in some third element $H$?"
It was answered in the negative (the counterexample seems to be due to Dixmier and Makar-Limanov).
Now replace the field of characteristic zero $k$ by a non-normal integral domain of characteristic zero $R$, for example $R=mathbb{C}[t^2,t^3]$.
Is it possible to adjust $P$ and $Q$ to some $tilde{P}$ and $tilde{Q}$ satisfying the following three conditions:
(i) $[tilde{P},tilde{Q}]=0$.
(ii) $tilde{P}$ and $tilde{Q}$ are not polynomials in some third element $H$.
(iii) $tilde{P}$ has a 'Dixmier mate', namely, there exists $B$ in the first Weyl algebra over $R$ such that $[tilde{P},B]=1$.
Remarks:
(1) The analogous question in $R[x,y]$ can be found here.
(2) In contrast, in $k[x,y]$, it is true that if $operatorname{Jac}(p,q)=0$, then $p=u(h)$ and $q=v(h)$, for some $h in k[x,y]$, $u(T),v(T) in k[T]$; see, for example, Corollary 1.3 in Nagata's paper ($k$ is an algebraically closed field of characteristic zero) or this question ($k$ is an arbitrary field).
$k$ can be replaced by any normal integral domain of characteristic zero, see this answer.
Any hints and comments are welcome!
ring-theory noncommutative-algebra
asked Nov 29 at 15:12
user237522
2,0871617
I have asked this question in MO, but have not received any comments/answers, so now I ask it here:
This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ are commuting elements of the first Weyl algebra. Is it true that $P$ and $Q$ are polynomials in some third element $H$?"
It was answered in the negative (the counterexample seems to be due to Dixmier and Makar-Limanov).
Now replace the field of characteristic zero $k$ by a non-normal integral domain of characteristic zero $R$, for example $R=mathbb{C}[t^2,t^3]$.
Is it possible to adjust $P$ and $Q$ to some $tilde{P}$ and $tilde{Q}$ satisfying the following three conditions:
(i) $[tilde{P},tilde{Q}]=0$.
(ii) $tilde{P}$ and $tilde{Q}$ are not polynomials in some third element $H$.
(iii) $tilde{P}$ has a 'Dixmier mate', namely, there exists $B$ in the first Weyl algebra over $R$ such that $[tilde{P},B]=1$.
Remarks:
(1) The analogous question in $R[x,y]$ can be found here.
(2) In contrast, in $k[x,y]$, it is true that if $operatorname{Jac}(p,q)=0$, then $p=u(h)$ and $q=v(h)$, for some $h in k[x,y]$, $u(T),v(T) in k[T]$; see, for example, Corollary 1.3 in Nagata's paper ($k$ is an algebraically closed field of characteristic zero) or this question ($k$ is an arbitrary field).
$k$ can be replaced by any normal integral domain of characteristic zero, see this answer.
Any hints and comments are welcome!
ring-theory noncommutative-algebra
ring-theory noncommutative-algebra
asked Nov 29 at 15:12
user237522
2,0871617
asked Nov 29 at 15:12
user237522
2,0871617
edited Nov 29 at 15:28
asked Nov 29 at 15:12
user237522
2,0871617
asked Nov 29 at 15:12
user237522
2,0871617
asked Nov 29 at 15:12
user237522
2,0871617
2,0871617
add a comment |
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