Editing Weyl algebra (section)

== Constructions ==
The Weyl algebras have different constructions, with different levels of abstraction.

=== Representation ===
The Weyl algebra <math>A_n</math> can be concretely constructed as a [[Algebra representation|representation]].

In the differential operator representation, similar to Schrödinger's canonical quantization, let <math>q_j</math> be represented by multiplication on the left by <math>x_j</math>, and let <math>p_j</math> be represented by differentiation on the left by <math>\partial_{x_j}</math>.

In the matrix representation, similar to the [[matrix mechanics]], <math>
A_1
</math> is represented by{{sfn|Coutinho|1997|pp=598–599}}<math display="block">
P=\begin{bmatrix}
0 & 1 & 0 & 0 & \cdots \\
0 & 0 & 2 & 0 & \cdots \\
0 & 0 & 0 & 3 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{bmatrix}, \quad Q=\begin{bmatrix}
0 & 0 & 0 & 0 & \ldots \\
1 & 0 & 0 & 0 & \cdots \\
0 & 1 & 0 & 0 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{bmatrix}
</math>

=== Generator ===
<math> A_n</math> can be constructed as a quotient of a free algebra in terms of generators and relations.
One construction starts with an abstract [[vector space]] ''V'' (of dimension 2''n'') equipped with a [[symplectic form]] ''ω''. Define the Weyl algebra ''W''(''V'') to be
: <math>W(V) := T(V) / (\!( v \otimes u - u \otimes v - \omega(v,u), \text{ for } v,u \in V )\!),</math>
where ''T''(''V'') is the [[tensor algebra]] on ''V'', and  the notation <math>(\!( )\!)</math> means "the [[ideal (ring theory)|ideal]] generated by".

In other words, ''W''(''V'') is the algebra generated by ''V'' subject only to the relation {{math|''vu'' − ''uv'' {{=}} ''ω''(''v'', ''u'')}}. Then, ''W''(''V'') is isomorphic to ''A<sub>n</sub>'' via the choice of a Darboux basis for {{mvar|ω}}.

<math> A_n</math> is also a [[quotient ring|quotient]] of the [[universal enveloping algebra]] of the [[Heisenberg algebra]], the [[Lie algebra]] of the [[Heisenberg group]], by setting the central element of the Heisenberg algebra (namely [''q'', ''p'']) equal to the unit of the universal enveloping algebra (called 1 above).

=== Quantization ===
The algebra ''W''(''V'') is a [[quantization (physics)|quantization]] of the [[symmetric algebra]] Sym(''V'').  If ''V'' is over a field of characteristic zero, then ''W''(''V'') is naturally isomorphic to the underlying vector space of the [[symmetric algebra]] Sym(''V'') equipped with a deformed product – called the Groenewold–[[Moyal product]] (considering the symmetric algebra to be polynomial functions on ''V''<sup>∗</sup>, where the variables span the vector space ''V'', and replacing ''iħ''  in the Moyal product formula with 1).

The isomorphism is given by the symmetrization map from Sym(''V'') to ''W''(''V'')
: <math>a_1 \cdots a_n \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(n)}~.</math>

If one prefers to have the ''iħ'' and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by ''q''<sub>''i''</sub> and ''iħ∂<sub>q<sub>i</sub></sub>''  (as per [[quantum mechanics]] usage).

Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the [[Moyal product|Moyal quantization]] (if for the latter one restricts to polynomial functions), but the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication.

Stated in another way, let the [[Moyal product|Moyal star product]] be denoted <math>f \star g</math>, then the Weyl algebra is isomorphic to <math>(\mathbb C[x_1, \dots, x_n], \star)</math>.{{sfn|Coutinho|1997|pp=602–603}}

In the case of [[exterior algebra]]s, the analogous quantization to the Weyl one is the [[Clifford algebra]], which is also referred to as the ''orthogonal Clifford algebra''.{{sfn|Lounesto|Ablamowicz|2004|p=xvi}}{{sfn|Micali|Boudet|Helmstetter|1992|pp=83-96}}

The Weyl algebra is also referred to as the '''symplectic Clifford algebra'''.{{sfn | Lounesto | Ablamowicz | 2004|p=xvi}}{{sfn | Micali | Boudet | Helmstetter | 1992 | pp=83-96}}{{sfn | Helmstetter | Micali | 2008 | p=xii}} Weyl algebras represent for symplectic [[bilinear form]]s the same structure that [[Clifford algebra]]s represent for non-degenerate symmetric bilinear forms.{{sfn | Helmstetter | Micali | 2008 | p=xii}}

=== D-module ===
The Weyl algebra can be constructed as a [[D-module]].{{sfn | Coutinho | 1997 | pp=600–601}} Specifically, the Weyl algebra corresponding to the polynomial ring <math>R[x_1, ..., x_n]</math> with its usual partial differential structure is precisely equal to Grothendieck's ring of differential operations <math>D_{\mathbb{A}^n_R / R}</math>.{{sfn | Coutinho | 1997 | pp=600–601}}

More generally, let <math>X</math> be a smooth scheme over a ring <math>R</math>. Locally, <math>X \to R</math> factors as an étale cover over some <math>\mathbb{A}^n_R</math> equipped with the standard projection.<ref>{{Cite web |title=Section 41.13 (039P): Étale and smooth morphisms—The Stacks project |url=https://stacks.math.columbia.edu/tag/039P |access-date=2024-09-29 |website=stacks.math.columbia.edu}}</ref> Because "''étale''" means "(flat and) possessing null cotangent sheaf",<ref>{{Cite web |title=etale morphism of schemes in nLab |url=https://ncatlab.org/nlab/show/etale+morphism+of+schemes |access-date=2024-09-29 |website=ncatlab.org}}</ref> this means that every D-module over such a scheme can be thought of locally as a module over the <math>n^\text{th}</math> Weyl algebra.

Let <math>R</math> be a [[commutative algebra]] over a subring <math>S</math>. The '''ring of differential operators''' <math>D_{R/S}</math> (notated <math>D_R</math> when <math>S</math> is clear from context) is inductively defined as a graded subalgebra of <math>\operatorname{End}_{S}(R)</math>:

* <math>D^0_R=R</math>
* <math>
D^k_R=\left\{d \in \operatorname{End}_{S}(R):[d, a] \in D^{k-1}_R \text { for all } a \in R\right\} .
</math>

Let <math>D_R</math> be the union of all <math>D^k_R</math> for <math>k \geq 0</math>. This is a subalgebra of <math>\operatorname{End}_{S}(R)</math>.

In the case <math>R = S[x_1, ..., x_n]</math>, the ring of differential operators of order <math>\leq n</math> presents similarly as in the special case <math>S = \mathbb{C}</math> but for the added consideration of "divided power operators"; these are operators corresponding to those in the complex case which stabilize <math>\mathbb{Z}[x_1, ..., x_n]</math>, but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit <math>D_{\mathbb{A}^n_\mathbb{Z} / \mathbb{Z}}</math>. One such example is the operator <math>\partial_{x_1}^{[p]} : x_1^N \mapsto {N \choose p} x_1^{N-p}</math>.

Explicitly, a presentation is given by

:<math>D_{S[x_1, \dots, x_\ell]/S}^n = S \langle x_1, \dots, x_\ell, \{\partial_{x_i}, \partial_{x_i}^{[2]}, \dots, \partial_{x_i}^{[n]}\}_{1 \leq i \leq \ell} \rangle</math>
with the relations
:<math>[x_i, x_j] = [\partial_{x_i}^{[k]}, \partial_{x_j}^{[m]}] = 0</math>
:<math>[\partial_{x_i}^{[k]}, x_j] = \left \{ \begin{matrix}\partial_{x_i}^{[k-1]} & \text{if }i=j \\ 0 & \text{if } i \neq j\end{matrix}\right.</math>
:<math>\partial_{x_i}^{[k]} \partial_{x_i}^{[m]} = {k+m \choose k} \partial_{x_i}^{[k+m]} ~~~~~\text{when }k+m \leq n</math>
where <math>\partial_{x_i}^{[0]} = 1</math> by convention. The Weyl algebra then consists of the limit of these algebras as <math>n \to \infty</math>.<ref>{{Cite journal |last=Grothendieck |first=Alexander |date=1964 |title=Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Première partie |url=http://www.numdam.org/item/PMIHES_1964__20__5_0/ |journal=Publications Mathématiques de l'IHÉS |language=en |volume=20 |pages=5–259 |issn=1618-1913}}</ref>{{Pg|location=Ch. IV.16.II}}

When <math>S</math> is a field of characteristic 0, then <math>D^1_R</math> is generated, as an <math>R</math>-module, by 1 and the <math>S</math>-[[Derivation (differential algebra)|derivations]] of <math>R</math>. Moreover, <math>D_R</math> is generated as a ring by the <math>R</math>-subalgebra <math>D^1_R</math>. In particular, if <math>S = \mathbb{C}</math> and <math>R=\mathbb{C}[x_1, ..., x_n]</math>, then <math>D^1_R=R+ \sum_i R \partial_{x_i} </math>. As mentioned, <math>A_n = D_R</math>.{{sfn | Coutinho | 1995 | pp=20-24}}