All Questions
Tagged with noncommutative-algebra weyl-algebra
8 questions with no upvoted or accepted answers
5
votes
0
answers
200
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A non-commutative analog of a result concerning a Jacobian pair
Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$.
Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$.
Similarly, define $t_y(E)$ to be the maximum among $...
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4
votes
0
answers
80
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Indecomposable injectives over Weyl algebras
Let $A=A_n(\mathbb{C})$ be the $n$-th Weyl algebra over the complex field. Then $A$ is a left Noetherian noncommutative ring. Is there a complete classification of indecomposable injective $A$-modules?...
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4
votes
0
answers
87
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Is $x \in A_1$ left algebraic over the subalgebra generated by $p$ and $q$, $[q,p]=1$?
Let $A_1:=A_1(x,y,k)$ be the first Weyl algebra over a field $k$ of characteristic zero,
namely, the $k$-algebra generated by $x$ and $y$ with relation $yx-xy=1$.
Let $f:(x,y) \mapsto (p,q)$ be a $k$-...
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4
votes
0
answers
152
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Nilpotent operator of the Weyl algebra
For a research project I'm currently working on, I came across the following problem:
Let $A=$ $^{k <x,y> }\Big/_{(yx-xy-1)}$ be the Weyl Algebra over a field $k$ of characteristic $p$, where $...
- 499
3
votes
0
answers
115
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The group of automorphisms and anti-automorphisms of the first Weyl algebra
Let $k$ be a field of characteristic zero, and let $A_1=A_1(k)$ be the first Weyl algebra.
It is well known (first proved by Dixmier, if I am not wrong) that the group of automorphisms of $A_1$, ...
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2
votes
0
answers
42
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Concerning $(x,y) \mapsto (x^{\frac{n}{r}+1}y + A,\mu x^{-\frac{n}{r}}+B)$
Let $r \in \mathbb{N}-\{0\}$.
Commutative case:
Let $f : (x,y) \mapsto (p,q)$ be a map from $\mathbb{C}[x,y]$ to $\mathbb{C}[x^{1/r},x^{-1/r},y]$ satisfying the following two conditions:
(i) $\...
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1
vote
0
answers
60
views
A variation on Dixmier's counterexample concerning centralizers in $A_1$
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 ...
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1
vote
0
answers
62
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Invertibility under base change for the Weyl algebra instead of for the polynomial algebra
From Lemma 1.1.8, we obtain the following:
Assume that $R \subseteq S$ are commutative rings
and
$u: R[x,y] \to R[x,y]$ is an $R$-algebra endomorphism
that has an invertible Jacobian, namely,
$Jac(u(x)...
- 2,837