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4 votes
0 answers
80 views

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?...
7 votes
1 answer
497 views

Weyl algebra as an Azumaya algebra over its centre

Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple ...
3 votes
1 answer
197 views

Gelfand-Kirillov dimension of the first Weyl algebra

How can we compute the Gelfand-Kirillov dimension (GK for short) of the first Weyl algebra? As we know we can look at the Weyl algebra as a generalized Weyl algebra in the following way: Let $A=\...
5 votes
2 answers
340 views

Gelfand-Kirillov dimension of generalized Weyl algebras

I believe that the Gelfand-Kirillov (GK) dimension for a generalized Weyl algebra $D(\sigma,a)$ is just the GKdim$(D) + 1$. Does anyone have a reference for this? I can find partial results, and I ...
2 votes
0 answers
42 views

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) $\...
5 votes
0 answers
200 views

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 $...
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 ...
1 vote
0 answers
62 views

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)...
4 votes
0 answers
87 views

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 $k$-...
5 votes
2 answers
243 views

A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero

Let $f,g \in \mathbb{C}[x,y]$. There is a well-known result, that can be found for example here, pages 19-20, that says the following: $f,g$ are algebraically dependent over $\mathbb{C}$ if and ...
3 votes
0 answers
115 views

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 $A_1$, ...
0 votes
1 answer
63 views

What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$?

Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$. What are ...
4 votes
1 answer
242 views

Locally nilpotent operators of the Weyl algebra

$\newcommand{\ad}{\operatorname{ad}}$As my recent post (here) did not receive any answers yet, I thought I would ask a similar question in which I'm also interested. Let $A=$ $^{k \langle x,y\rangle }...
4 votes
0 answers
152 views

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 $...
2 votes
1 answer
341 views

Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?
7 votes
2 answers
489 views

How big can a commutative subalgebra of Weyl algebra be?

Consider the smallest Weyl algebra $A_1=\{q,p; qp-pq=1\}$. It is known that there exist pairs of commuting elements, say $L$ and $M$, that obey various polynomial relations, e.g. elliptic curves. I ...
9 votes
5 answers
2k views

Proof a Weyl Algebra isn't isomorphic to a matrix ring over a division ring

Can anyone prove that a Weyl Algebra is not isomorphic to a matrix ring over a division ring?
7 votes
1 answer
567 views

Depth Zero Ideals in the Homogenized Weyl Algebra

Let $\mathcal{D}$ be the $n$th Weyl algebra $ \mathcal{D} :=k[x_1,...,x_n,\partial_1,...,\partial_n] $, where $\partial_ix_i-x_i\partial_i=1$. Let $\widetilde{\mathcal{D}}$ be its Rees algebra, ...