Editing Weyl algebra

{{Short description|Differential algebra}}
In [[abstract algebra]], the '''Weyl algebras''' are abstracted from the [[ring (mathematics)|ring]] of [[differential operator]]s with [[polynomial]] coefficients. They are named after [[Hermann Weyl]], who introduced them to study the [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]] in [[quantum mechanics]]. 

In the simplest case, these are differential operators. Let <math>F</math> be a [[field (mathematics)|field]], and let <math>F[x]</math> be the [[polynomial ring|ring of polynomials]] in one variable with coefficients in <math>F</math>. Then the corresponding Weyl algebra consists of differential operators of form

: <math> f_m(x) \partial_x^m + f_{m-1}(x) \partial_x^{m-1} + \cdots + f_1(x) \partial_x + f_0(x) </math>

This is the '''first Weyl algebra''' <math>A_1</math>. The '''''n''-th Weyl algebra''' <math>A_n</math> are constructed similarly. 

Alternatively, <math>A_1</math> can be constructed as the [[quotient ring|quotient]] of the [[free algebra]] on two generators, ''q'' and ''p'', by the [[ideal (ring theory)|ideal]] generated by <math>([p,q] - 1)</math>. Similarly, <math>A_n</math> is obtained by quotienting the free algebra on ''2n'' generators by the ideal generated by<math display="block"> ([p_i,q_j] - \delta_{i,j}), \quad \forall i, j = 1, \dots, n</math>where <math> \delta_{i,j}</math> is the [[Kronecker delta]].

More generally, let <math> (R,\Delta) </math> be a partial [[differential ring]] with commuting derivatives <math> \Delta = \lbrace \partial_1,\ldots,\partial_m \rbrace </math>. The '''Weyl algebra associated to''' <math>(R,\Delta)</math> is the noncommutative ring <math> R[\partial_1,\ldots,\partial_m] </math> satisfying the relations <math> \partial_i r = r\partial_i + \partial_i(r) </math> for all <math> r \in R </math>. The previous case is the special case where <math> R=F[x_1,\ldots,x_n] </math> and <math> \Delta = \lbrace \partial_{x_1},\ldots,\partial_{x_n} \rbrace </math> where <math> F </math> is a field.

This article discusses only the case of <math>A_n</math> with underlying field <math>F</math> [[Characteristic (algebra)|characteristic zero]], unless otherwise stated.

The Weyl algebra is an example of a [[simple ring]] that is not a [[matrix ring]] over a [[division ring]]. It is also a noncommutative example of a [[domain (ring theory)|domain]], and an example of an [[Ore extension]].

== Motivation ==
{{see also|Canonical commutation relation}}
The Weyl algebra arises naturally in the context of [[quantum mechanics]] and the process of [[canonical quantization]]. Consider a classical [[phase space]] with canonical coordinates <math>(q_1, p_1, \dots, q_n, p_n) </math>.  These coordinates satisfy the [[Poisson bracket]] relations:<math display="block">
\{q_i, q_j\} = 0, \quad \{p_i, p_j\} = 0, \quad \{q_i, p_j\} = \delta_{ij}.
</math>In canonical quantization, one seeks to construct a [[Hilbert space]] of states and represent the classical observables (functions on phase space) as [[Self-adjoint operator|self-adjoint operators]] on this space. The canonical commutation relations are imposed:<math display="block">
[\hat{q}_i, \hat{q}_j] = 0, \quad [\hat{p}_i, \hat{p}_j] = 0, \quad [\hat{q}_i, \hat{p}_j] = i\hbar \delta_{ij},
</math>where <math>[\cdot, \cdot]</math> denotes the [[commutator]]. Here, <math>\hat{q}_i</math> and <math>\hat{p}_i</math> are the operators corresponding to <math>q_i</math> and <math>p_i</math> respectively. [[Erwin Schrödinger]] proposed in 1926 the following:{{sfn | Landsman | 2007 | p=428}}

* <math>\hat{q_j}</math> with multiplication by <math>x_j</math>.
* <math>\hat{p}_j</math> with <math>-i\hbar \partial_{x_j}</math>.

With this identification, the canonical commutation relation holds.

== 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}}

== Properties of ''A<sub>n</sub>'' ==
Many properties of <math>
A_1
</math> apply to <math>
A_n
</math> with essentially similar proofs, since the different dimensions commute.

=== General Leibniz rule ===
{{Main|General Leibniz rule}}

{{Math theorem
| name = Theorem
| note = general Leibniz rule
| math_statement = <math display="block">
p^k q^m = \sum_{l=0}^k \binom{k}{l} \frac{m!}{(m-l)!} q^{m-l} p^{k-l} = q^mp^k + mk q^{m-1}p^{k-1} + \cdots 
</math>
}}

{{Math proof|title=Proof|proof=
 Under the <math>
p \mapsto x, q \mapsto \partial_x
</math> representation, this equation is obtained by the general Leibniz rule. Since the general Leibniz rule is provable by algebraic manipulation, it holds for <math>
A_1
</math> as well.
}}In particular, <math display="inline">[q, q^m p^n] = -nq^mp^{n-1}</math> and <math display="inline">[p, q^mp^n] = mq^{m-1}p^n</math>.
{{Math theorem
| math_statement = The [[Center (ring theory)|center]] of Weyl algebra <math>A_n</math> is the underlying field of constants <math>F</math>.
| name = Corollary
}}

{{Math proof|title=Proof|proof= 
If the commutator of <math>f</math> with either of <math>p, q</math> is zero, then by the previous statement, <math>f</math> has no monomial <math>p^nq^m</math> with <math>n > 0</math> or <math>m > 0</math>.
}}
=== Degree ===
{{Math theorem
| name = Theorem
| note = 
| math_statement = <math>
A_n
</math> has a basis <math>
\{q^m p^n : m, n \geq 0\}
</math>.{{Sfn|Coutinho|1995|p=9|loc=Proposition 2.1}}
}}

{{Math proof|title=Proof|proof=
 By repeating the commutator relations, any monomial can be equated to a linear sum of these. It remains to check that these are linearly independent. This can be checked in the differential operator representation. For any linear sum <math>
\sum_{m, n} c_{m,n} x^m \partial_x^n
</math> with nonzero coefficients, group it in descending order: <math>
p_N(x) \partial_x^N + p_{N-1}(x) \partial_x^{N-1} + \cdots + p_M(x) \partial_x^M
</math>, where <math>
p_M
</math> is a nonzero polynomial. This operator applied to <math>
x^M
</math> results in <math>
M! p_M(x) \neq 0
</math>.
}}

This allows <math>
A_1
</math> to be a [[graded algebra]], where the degree of <math>
\sum_{m, n} c_{m,n} q^m p^n
</math> is <math>
\max (m + n)
</math> among its nonzero monomials. The degree is similarly defined for <math>
A_n
</math>.

{{Math theorem
| name = Theorem
| math_statement = For <math>A_n</math>:{{sfn | Coutinho | 1995 | pp=14-15}}
 
* <math>
\deg(g + h) \leq \max(\deg(g), \deg(h)) 
</math>
* <math>
\deg([g, h]) \leq \deg(g) + \deg(h) - 2 
</math>
* <math>
\deg(g h) = \deg(g) + \deg(h) 
</math>
}}

{{Math proof|title=Proof|proof=
We prove it for <math>A_1</math>, as the <math>A_n</math> case is similar.

The first relation is by definition. The second relation is by the general Leibniz rule. For the third relation, note that <math>\deg(g h) \leq \deg(g) + \deg(h)</math>, so it is sufficient to check that <math>gh</math> contains at least one nonzero monomial that has degree <math>\deg(g) + \deg(h)</math>. To find such a monomial, pick the one in <math>g</math> with the highest degree. If there are multiple such monomials, pick the one with the highest power in <math>q</math>. Similarly for <math>h</math>. These two monomials, when multiplied together, create a unique monomial among all monomials of <math>gh</math>, and so it remains nonzero.
}}

{{Math theorem
| name = Theorem
| math_statement = <math>A_n</math> is a [[Simple algebra|simple]] [[Domain (ring theory)|domain]].{{sfn | Coutinho | 1995 | p=16}}
}}
That is, it has no [[Ideal (ring theory)|two-sided nontrivial ideals]] and has no [[zero divisor]]s.
{{Math proof|title=Proof|proof=
Because <math>\deg(gh) = \deg(g) + \deg(h)</math>, it has no zero divisors.

Suppose for contradiction that <math>I</math> is a nonzero two-sided ideal of <math>A_1</math>, with <math>I \neq A_1</math>. Pick a nonzero element <math>f \in I</math> with the lowest degree. 

If <math>f</math> contains some nonzero monomial of form <math>xx^m\partial^n = x^{m+1} \partial^n</math>, then 
<math display=block>
[\partial, f] = \partial f - f \partial
</math>
contains a nonzero monomial of form 
<math display=block>
\partial x^{m+1} \partial^n -  x^{m+1} \partial^n \partial = (m+1) x^m \partial^n.
</math>
Thus <math>[\partial, f]</math> is nonzero, and has degree <math>\leq \deg(f)-1</math>. As <math>I</math> is a two-sided ideal, we have <math>[\partial, f] \in I</math>, which contradicts the minimality of <math>\deg(f)</math>.

Similarly, if <math>f</math> contains some nonzero monomial of form <math>x^m\partial^n\partial</math>, then <math>[x, f] = xf - fx</math> is nonzero with lower degree.
}}

=== Derivation ===
{{See|Derivation (differential algebra)}}{{Math theorem|
| math_statement = The derivations of <math display="inline">A_n</math> are in bijection with the elements of <math display="inline">A_n</math> up to an additive scalar.{{sfn | Dirac | 1926 | pp=415–417}}}}
That is, any derivation <math display="inline">D</math> is equal to <math display="inline">[\cdot, f]</math> for some <math display="inline">f \in A_n</math>; any <math display="inline">f\in A_n</math> yields a derivation <math display="inline">[\cdot, f]</math>; if <math display="inline">f, f' \in A_n</math> satisfies <math display="inline">[\cdot, f] = [\cdot, f']</math>, then <math display="inline">f - f' \in F</math>.

The proof is similar to computing the potential function for a conservative polynomial vector field on the plane.{{sfn | Coutinho | 1997 | p=597}}

{{Collapse top|title=Proof}}
Since the commutator is a derivation in both of its entries, <math display="inline">[\cdot, f]</math> is a derivation for any <math display="inline">f\in A_n</math>. Uniqueness up to additive scalar is because the center of <math display="inline">A_n</math> is the ring of scalars.

It remains to prove that any derivation is an inner derivation by induction on <math display="inline">n</math>.

Base case: Let <math display="inline">D: A_1 \to A_1</math> be a linear map that is a derivation. We construct an element <math display="inline">r</math> such that <math display="inline">[p, r] = D(p), [q,r] = D(q)</math>. Since both <math display="inline">D</math> and <math display="inline">[\cdot, r]</math> are derivations, these two relations generate <math display="inline">[g, r] = D(g)</math> for all <math display="inline">g\in A_1</math>.

Since <math display="inline">[p, q^mp^n] = mq^{m-1}p^n</math>, there exists an element <math display="inline">f = \sum_{m,n} c_{m,n} q^m p^n</math> such that <math display="block">
      [p, f] =  \sum_{m,n} m c_{m,n} q^m p^n = D(p)
      </math> 

<math display="block">
      \begin{aligned}
      0 &\stackrel{[p, q] = 1}{=} D([p, q]) \\
      &\stackrel{D \text{ is a derivation}}{=} [p, D(q)] + [D(p), q] \\
      &\stackrel{[p,f] = D(p)}{=}  [p, D(q)] + [[p,f], q] \\
      &\stackrel{\text{Jacobi identity}}{=} [p, D(q) - [q, f]]
      \end{aligned}
      </math> 

Thus, <math display="inline">D(q) = g(p) + [q, f]</math> for some polynomial <math display="inline">g</math>. Now, since <math display="inline">[q, q^m p^n] = -nq^mp^{n-1}</math>, there exists some polynomial <math display="inline">h(p)</math> such that <math display="inline">[q, h(p)] = g(p)</math>. Since <math display="inline">[p, h(p)] = 0</math>, <math display="inline">r = f + h(p)</math> is the desired element.

For the induction step, similarly to the above calculation, there exists some element <math display="inline">r \in A_n</math> such that <math display="inline">[q_1, r] = D(q_1), [p_1, r] = D(p_1)</math>.

Similar to the above calculation, <math display="block">
 [x, D(y) - [y, r]] = 0
 </math> for all <math display="inline">x \in \{p_1, q_1\}, y \in \{p_2, \dots, p_n, q_2, \dots, q_n\}</math>. Since <math display="inline">[x, D(y) - [y, r]]</math> is a derivation in both <math display="inline">x</math> and <math display="inline">y</math>, <math display="inline">[x, D(y) - [y, r]] = 0</math> for all <math display="inline">x\in \langle p_1, q_1\rangle</math> and all <math display="inline">y \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math>. Here, <math display="inline">\langle \rangle</math> means the subalgebra generated by the elements.

Thus, <math display="inline">\forall y \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math>, <math display="block">
 D(y) - [y, r] \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle
 </math>

Since <math display="inline">D - [\cdot, r]</math> is also a derivation, by induction, there exists <math display="inline">r' \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math> such that <math display="inline">D(y) - [y, r] = [y, r']</math> for all <math display="inline">y \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math>.

Since <math display="inline">p_1, q_1</math> commutes with <math display="inline">\langle p_2, \dots, p_n, q_2, \dots, q_n\rangle</math>, we have <math display="inline">D(y) = [y, r + r']</math> for all <math>y \in \{p_1, \dots, p_n, q_1, \dots, q_n\}</math>, and so for all of <math>A_n</math>.
{{Collapse bottom}}

== Representation theory ==
{{further|Stone–von Neumann theorem}}

=== Zero characteristic ===
In the case that the ground field {{mvar|F}} has characteristic zero, the ''n''th Weyl algebra is a [[simple ring|simple]] [[Noetherian ring|Noetherian]] [[domain (ring theory)|domain]].{{sfn | Coutinho | 1995 | p=70}}  It has [[global dimension]] ''n'', in contrast to the ring it deforms, Sym(''V''), which has global dimension 2''n''.

It has no finite-dimensional representations. Although this follows from simplicity, it can be more directly shown by taking the trace of ''σ''(''q'') and ''σ''(''Y'') for some finite-dimensional representation ''σ'' (where {{nowrap|1=[''q'',''p''] = 1}}).
: <math> \mathrm{tr}([\sigma(q),\sigma(Y)])=\mathrm{tr}(1)~.</math>
Since the trace of a commutator is zero, and the trace of the identity is the dimension of the representation, the representation must be zero dimensional.

In fact, there are stronger statements than the absence of finite-dimensional representations.  To any finitely generated ''A<sub>n</sub>''-module ''M'', there is a corresponding subvariety Char(''M'') of {{nowrap|''V'' × ''V''<sup>∗</sup>}} called the 'characteristic variety'{{what|date=August 2016}} whose size roughly corresponds to the size{{what|date=August 2016}} of ''M'' (a finite-dimensional module would have zero-dimensional characteristic variety).  Then [[Bernstein's inequality (mathematical analysis)|Bernstein's inequality]] states that for ''M'' non-zero,
: <math>\dim(\operatorname{char}(M))\geq n</math>
An even stronger statement is [[Gabber's theorem]], which states that Char(''M'') is a [[Lagrangian submanifold|co-isotropic]] subvariety of {{nowrap|''V'' × ''V''<sup>∗</sup>}} for the natural symplectic form.

=== Positive characteristic ===
The situation is considerably different in the case of a Weyl algebra over a field of [[characteristic (algebra)|characteristic]] {{nowrap|''p'' > 0}}.

In this case, for any element ''D'' of the Weyl algebra, the element ''D<sup>p</sup>'' is central, and so the Weyl algebra has a very large center.  In fact, it is a finitely generated module over its center; even more so, it is an [[Azumaya algebra]] over its center.  As a consequence, there are many finite-dimensional representations which are all built out of simple representations of dimension ''p''.

== Generalizations ==
The [[Ideal (ring theory)|ideals]] and automorphisms of <math>A_1</math> have been well-studied.{{sfn|Berest|Wilson|2000|pp=127–147}}{{sfn|Cannings|Holland|1994|pp=116–141}} The [[moduli space]] for its right ideal is known.{{sfn|Lebruyn|1995|pp=32–48}} However, the case for <math>A_n</math> is considerably harder and is related to the [[Jacobian conjecture]].{{sfn|Coutinho|1995|loc=section 4.4}}

For more details about this quantization in the case ''n'' = 1 (and an extension using the [[Fourier transform]] to a class of integrable functions larger than the polynomial functions), see [[Wigner–Weyl transform]].

Weyl algebras and Clifford algebras admit a further structure of a [[*-algebra]], and can be unified as even and odd terms of a [[superalgebra]], as discussed in [[CCR and CAR algebras]].

=== Affine varieties ===
Weyl algebras also generalize in the case of algebraic varieties. Consider a polynomial ring
: <math>R = \frac{\mathbb{C}[x_1,\ldots,x_n]}{I}.</math>
Then a differential operator is defined as a composition of <math>\mathbb{C}</math>-linear derivations of <math>R</math>. This can be described explicitly as the quotient ring
: <math> \text{Diff}(R) = \frac{\{ D \in A_n\colon D(I) \subseteq I \}}{ I\cdot A_n}.</math>

== See also ==
* [[Jacobian conjecture]]
* [[Dixmier conjecture]]

== Notes == 
{{reflist}}
== References ==
* {{cite book | last=Coutinho | first=S. C. | title=A Primer of Algebraic D-Modules | publisher=Cambridge University Press | publication-place=Cambridge [England] ; New York, NY, USA | date=1995| isbn=978-0-521-55119-9|doi=10.1017/cbo9780511623653}}
* {{cite journal | last=Coutinho | first=S. C. | title=The Many Avatars of a Simple Algebra | journal=The American Mathematical Monthly | volume=104 | issue=7 | date=1997 | issn=0002-9890 | doi=10.1080/00029890.1997.11990687 | pages=593–604}}
* {{cite journal | last=Dirac | first=P. A. M. | authorlink=Paul Dirac | title=On Quantum Algebra | journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] | volume=23 | issue=4 | date=1926 | issn=0305-0041 | doi=10.1017/S0305004100015231 | pages=412–418}}
* {{cite book | last=Helmstetter | first=J. | last2=Micali | first2=A. | title=Quadratic Mappings and Clifford Algebras | publisher=Birkhäuser | publication-place=Basel ; Boston | date=2008 | isbn=978-3-7643-8605-4 | oclc=175285188}}
* {{cite book | last=Landsman | first=N.P. | title=Philosophy of Physics | chapter=BETWEEN CLASSICAL AND QUANTUM | publisher=Elsevier | date=2007 | isbn=978-0-444-51560-5 | doi=10.1016/b978-044451560-5/50008-7}}
* {{cite book | last=Lounesto | first=P. | last2=Ablamowicz | first2=R. | title=Clifford Algebras | publisher=Springer Science & Business Media | publication-place=Boston | date=2004 | isbn=0-8176-3525-4}}
* {{cite book | last=Micali | first=A. | last2=Boudet | first2=R. | last3=Helmstetter | first3=J. | title=Clifford Algebras and their Applications in Mathematical Physics | publisher=Springer Science & Business Media | publication-place=Dordrecht | date=1992 | isbn=0-7923-1623-1}}
* {{cite journal |first1=M. Rausch |last1=de Traubenberg |first2=M. J. |last2=Slupinski |first3=A. |last3=Tanasa |title=Finite-dimensional Lie subalgebras of the Weyl algebra |journal=J. Lie Theory |year=2006  |volume=16 |pages=427–454 |arxiv=math/0504224}} 
* {{cite book |first=Will |last=Traves |chapter=Differential Operations on Grassmann Varieties |editor-last=Campbell |editor-first=H. |editor2-last=Helminck |editor2-first=A. |editor3-last=Kraft |editor3-first=H. |editor4-last=Wehlau |editor4-first=D. |title=Symmetry and Spaces |publisher=Birkhäuse |series=Progress in Mathematics |volume=278 |year=2010 |isbn=978-0-8176-4875-6 |pages=197–207 |doi=10.1007/978-0-8176-4875-6_10 }}
* {{cite book |author-link=Tsit Yuen Lam |author=Tsit Yuen Lam |title=A first course in noncommutative rings |publisher=Springer |edition=2nd |year=2001 |isbn=978-0-387-95325-0 |pages=6 |volume=131 |series=[[Graduate Texts in Mathematics]]}}
* {{cite journal |last=Berest |first=Yuri |last2=Wilson |first2=George |date=September 1, 2000 |title=Automorphisms and ideals of the Weyl algebra |journal=Mathematische Annalen |volume=318 |issue=1 |pages=127–147 |doi=10.1007/s002080000115 |issn=0025-5831|arxiv=math/0102190 }}
* {{cite journal |last=Cannings |first=R.C. |last2=Holland |first2=M.P. |year=1994 |title=Right Ideals of Rings of Differential Operators |journal=Journal of Algebra |publisher=Elsevier BV |volume=167 |issue=1 |pages=116–141 |doi=10.1006/jabr.1994.1179 |issn=0021-8693}}
* {{cite journal |last=Lebruyn |first=L. |year=1995 |title=Moduli Spaces for Right Ideals of the Weyl Algebra |journal=Journal of Algebra |publisher=Elsevier BV |volume=172 |issue=1 |pages=32–48 |doi=10.1006/jabr.1995.1046 |issn=0021-8693|hdl=10067/123950151162165141 |hdl-access=free }}

[[Category:Algebras]]
[[Category:Differential operators]]
[[Category:Ring theory]]