# Questions tagged [weyl-algebra]

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### Can the Weyl algebra be free over its invariant subalgebra?

Let $k$ be an algebraically closed field of zero characteristic, let $P_n$ denote the polynomial algebra in $n$ indeterminates, and let $G$ be a finite group of linear automorphisms. Then, by ...
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### Interplay beween simplicial and Weyl algebra identities

Recall that the (first) Weyl algebra is the algebra generated by $x,y$ with the relation $xy-yx=1$. It can be realized as the algebra of differential operators on $k[x]$, where one generator acts as ...
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### Shift Operators and the Weyl Algebra

I have a question about the action of a shift operator $E$ on polynomials $Ep(x) = p(x+1)$ in the context of linear differential operators in one variable with polynomial coefficients, i.e. ...
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### 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 ...
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### Computing intersection of Weyl algebra ideal with certain subring

Let $D=k [x_1,\ldots, x_n, \partial_1,\ldots, \partial_n]$ be the nth Weyl algebra over the characteristic zero field $k$. Set $\theta_i=x_i\partial_i$. Let $I$ be a left ideal in $D$. Is there a ...
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### 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 ...
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### Computations in Weyl algebra with rational function coefficients

I am looking for a software to perform calculations with modules over the algebra $R_n=\mathbb{C}(x_1\ldots x_n)\langle \partial_1\ldots\partial_n\rangle$ of differential operators with rational ...
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### Computer algebra system for Weyl algebra computations

Does anyone have a suggestion for the best computer program to perform calculations in the 2nd Weyl algebra?
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### Supertrace on Weyl algebra

Consider Weyl algebra, i.e. the algebra of $x^i$ and $p_i=\frac{\partial}{\partial x^i}$, its elements are differential operators $F(x,p)$. Weyl algebra is $\mathbb{Z}_2$ graded, hence one ask if ...
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### 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?
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### The Weyl algebra modules which are also rings

Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl ...
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### 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 ...
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### Modules over rings of differential operators

If $M$ is a left $\mathbb{C}[t] \langle \partial _t \rangle$-module ( a left module over the Weyl algebra), then $\mathrm{Hom}(M,\mathbb{C}[t] \langle \partial _t \rangle)$ is equipped with a ...
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### Koszul duality between Weyl and Clifford algebras?

Koszul duality Given a finite-dimensional $k$-vector space $V$ (I am happy taking $k = \mathbb{C}$ anywhere in the following if it makes a difference) and a subspace $R \subseteq V \otimes V$, we can ...
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### A simple proof of the Weyl algebra's rigidity.

I am wondering if there is a nice presentation of the Hochschild cohomology of $A_n$ the Weyl algebra. It is known that $H^m(A_n,A_n)=0$ for $m>0$, and thus it is rigid. A proof can be found in ...
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### Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian)

The article http://arxiv.org/abs/hep-th/0206233 (published in Ann. of Math. (2) 161 (2005), no. 3) deals with linear differential operators $D$ for which there exists another linear differential ...
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### Does the image of a differential operator always contain an ideal?

Let $\delta$ denote a non-zero complex algebraic differential operator in a single variable x. That is, it can be written as a sum $$\delta = \sum_i f_i\partial_x^i$$ where there $f_i$ are complex ...
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### 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, ...
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