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Let  $  K $
be a commutative field and  $  n $
a positive integer. The ring of differential operators with coefficients in the polynomial ring  $  K [ x ] = K [ x _ {1} \dots x _ {n} ] $
is denoted by  $  A _ {n} ( K) $
and called the Weyl algebra in  $  n $
variables over  $  K $. 
Identifying  $  K[ x] $
with the subring of zero-order differential operators, it follows that the ring  $  A _ {n} ( K) $
is generated by  $  K[ x] $
and the derivation operators  $  \{ \partial  _ {i} = \partial  / \partial  x _ {i} \} _ {1}  ^ {n} $. 
The commutators  $  [ \partial  _ {i} , x _ {i} ] = 1 $
for every  $  i $. 
So  $  A _ {n} ( K) $
is a non-commutative ring. Every element has a unique representation

$$ 
P( x, \partial  )  =  \sum _ {v = 0 } ^ { m }  p _  \alpha  ( x) \partial   ^  \alpha  ,
$$

where  $  \partial   ^  \alpha  $
are monomials in the derivation operators. The largest integer  $  m $
such that a polynomial coefficient  $  p _  \alpha  ( x) $
is non-zero with  $  | \alpha | = m $
is the order of the differential operator  $  P $. 
The order yields a filtration (cf. [[Filtered module|Filtered module]]) and the associated graded ring (cf. [[Graded module|Graded module]])

$$ 
 \mathop{\rm gr} ( A _ {n} ( K))  =  \oplus _ {m \geq  0 }
 \mathop{\rm gr} _ {m} ( A _ {n} ( K) ),
$$

where  $   \mathop{\rm gr} _ {m} ( A _ {n} ( K) ) $
is the set of operators of order  $  m $, 
divided by those of order  $  m - 1 $
at most. It is well-known that the associated graded ring is isomorphic to the polynomial ring in  $  2n $
variables over  $  K $, 
where  $  \{ \sigma _ {0} ( x _ {\mathbf . }  ) , \sigma _ {1} ( \partial  _ {\mathbf . }  ) \} $
are the generators.

==Ring-theoretic properties.==
Here only the case when the field  $  K $
has characteristic zero is discussed. If  $   \mathop{\rm char} ( K) > 0 $, 
the results below are no longer valid. For material when  $   \mathop{\rm char} ( K) > 0 $
see [[#References|[a30]]]. From now on  $   \mathop{\rm char} ( K) = 0 $. 
Then  $  A _ {n} ( K) $
is a [[Simple ring|simple ring]] and since  $   \mathop{\rm gr} ( A _ {n} ( K)) $
is Noetherian and commutative, it follows that  $  A _ {n} ( K) $
is both left and right Noetherian. By [[#References|[a43]]], every left ideal of  $  A _ {n} ( K) $
is generated by two elements. The global [[Homological dimension|homological dimension]] of  $  A _ {n} ( K) $
is equal to  $  n $. 
This result was proved in [[#References|[a37]]]. The case  $  n = 1 $
was settled before in [[#References|[a35]]]. Another important result is the involutivity of characteristic ideals.

To explain this, one considers a finitely-generated left  $  A _ {n} ( K) $-
module  $  M $. 
A good filtration on  $  M $
consists of an increasing sequence of  $  K[ x] $-
submodules  $  \{ M _ {v} \} $
such that  $  \partial  _ {i} M _ {v} \subset  M _ {v+1} $
for all pairs  $  i, v $, 
and the associated graded module  $  \oplus M _ {v} / M _ {v-1} $
is finitely generated over  $   \mathop{\rm gr} ( A _ {n} ( K)) $. 
A module can be equipped with different good filtrations. But there exists a unique graded ideal of  $   \mathop{\rm gr} ( A _ {n} ( K)) $, 
given as the radical of the annihilating ideal of  $  \oplus M _ {v} / M _ {v-1} $
for any good filtration. It is denoted by  $  J( M) $
and called the characteristic ideal of  $  M $. 
On  $   \mathop{\rm gr} ( A _ {n} ( K)) $
there exists a Poisson product such that  $  \{ \sigma _ {1} ( \partial  _ {v} ) , \sigma _ {0} ( x _ {i} ) \} = \Delta _ {iv} $. 
The involutivity theorem asserts that

$$ \tag{a1 }
\{ J ( M) , J( M) \}  \subset   J( M)
$$

for every finitely-generated left  $  A _ {n} ( K) $-
module. In the special case when  $  K = \mathbf C $, 
the Nullstellen Satz (cf. [[Hilbert theorem|Hilbert theorem]]) identifies  $  J( M) $
with an algebraic set in the symplectic cotangent space  $  T  ^ {*} ( \mathbf C  ^ {n} ) $, 
denoted by  $   \mathop{\rm Char} ( M) $
and called the characteristic variety of  $  M $(
cf. also [[Characteristic manifold|Characteristic manifold]]). Then (a1) means that  $   \mathop{\rm Char} ( M) $
is involutive in the symplectic cotangent space.

The involutivity implies that the dimension of  $   \mathop{\rm gr} ( A _ {n} ( K)) / J( M) $
is at least  $  n $
when  $  M $
is a non-zero  $  A _ {n} ( K) $-
module. This can be used to prove that  $   \mathop{\rm gl}.dim ( A _ {n} ( K)) = n $, 
using the fact that  $  A _ {n} ( K) $
is a regular Auslander ring. See [[#References|[a8]]] for a survey of this. The result (a1) was proved by micro-local analysis in [[#References|[a40]]]. An algebraic proof was found later in [[#References|[a14]]]. In [[#References|[a26]]] characteristic ideals were used to show that if  $  W \subset   \mathop{\rm gr} ( A _ {n} ( K)) $
is a multiplicative set formed by homogeneous elements and  $  S $
is the multiplicative set in  $  A _ {n} ( K) $
whose principal symbols belong to  $  W $, 
then  $  S $
satisfies the two-sided Ore condition. So the universal  $  S $-
inverting ring is the two-sided ring of Ore fractions  $  S  ^ {-1} A _ {n} ( K) $. 
In particular, this applies when  $  S $
is the set of non-zero elements. The resulting division ring  $  D _ {n} ( K) $
is related to division rings of certain enveloping algebras divided by primitive ideals.

==Holonomic modules.==
The equality  $   \mathop{\rm gl}.dim ( A _ {n} ( K)) = n $
implies that its Krull dimension is at most  $  n $. 
The equality  $   \mathop{\rm Kr}.dim ( A _ {n} ( K)) = n $
was actually proved before the discoveries above in [[#References|[a15]]], using a trace formula which shows that every non-zero  $  A _ {1} ( K) $-
module is an infinite-dimensional vector space over  $  K $. 
Of special interest is the set of finitely-generated left or right  $  A _ {n} ( K) $-
modules  $  M $
such that  $   \mathop{\rm dim} (  \mathop{\rm gr} ( K) / J( M)) = n $. 
Such modules are called holonomic and enjoy finiteness properties, e.g. every holonomic module is Artinian. The converse is not true, since [[#References|[a43]]] gives examples of cyclic modules  $  M = A _ {n} ( K) / A _ {n} ( K) P $
which are simple. Here  $  n $
is any positive integer and  $  J ( M) $
is a principal ideal, so  $  M $
is non-holonomic when  $  n > 1 $.

An important class of holonomic modules arise as follows: If  $  P( x) \in K[ x] $, 
then the subring of rational functions  $  K[ x, P  ^ {-1} ] $
is a holonomic  $  A _ {n} ( K) $-
module. This was proved by J. Bernstein in [[#References|[a3]]], [[#References|[a4]]], in which also a functional equation was found expressed by the equality:

$$ \tag{a2 }
b( s) P( x)  ^ {s}  =  \sum s  ^ {v} Q _ {v} ( x, \partial  ) ( P( x)  ^ {s+1} ) .
$$

Here  $  b( s) $
is a polynomial, chosen to have smallest possible degree and highest coefficient one. It is called the Bernstein–Sato polynomial of  $  P $. 
The case when  $  K = \mathbf C $
is of particular interest. It is proved in [[#References|[a27]]] that the roots of  $  b( s) $
consist of strictly negative rational numbers. See also [[#References|[a6]]] for this. The roots of  $  b( s) $
are related to the monodromy acting in  $  X = \mathbf C  ^ {n} \setminus  P  ^ {-1} ( 0) $
under the mapping  $  P :  X \rightarrow \mathbf C  ^ {*} $, 
where it is assumed that zero is the only critical value of  $  P $. 
In [[#References|[a31]]] it is proved that the union  $  \{ {e ^ {- 2 \pi i \alpha } } : {\alpha \in b  ^ {-1} ( 0) } \} $
is equal to the union of eigenvalues of monodromy in every dimension in  $  \mathbf C  ^ {n} \setminus  P $. 
Bernstein's functional equation gives meromorphic continuations of distributions. The meromorphic  $  \mathfrak D \mathfrak b ( \mathbf C  ^ {n} ) $-
valued function  $  \mu _ {s} = \int | P |  ^ {2s} $
exists, with poles contained in the set  $  \{ {\cup ( \alpha - v ) } : {\alpha \in b  ^ {-1} ( 0)  \textrm{ and }  v \in \mathbf N } \} $. 
The roots of the  $  b $-
function give an effective contribution to the set of poles. Namely, for any root  $  \alpha $
of  $  b( s) $
there exists some  $  v \in \mathbf N $
such that  $  \alpha - v $
is a pole of  $  \mu _ {s} $. 
This is proved in [[#References|[a1]]].

==Fundamental solutions.==
Let  $  P( D) $
be a differential operator with constant coefficients. Using the Fourier transform, and replacing  $  \mathbf C  ^ {n} $
by  $  \mathbf R  ^ {n} $, 
the  $  \mathfrak D \mathfrak b ( \mathbf R  ^ {n} ) $-
valued function  $  \mu _ {s} $
exists, acting on test-forms  $  \phi ( x) $
by

$$ 
\langle  \mu _ {s} , \phi \rangle  =  \int\limits _ {\mathbf R  ^ {n} }
P( \xi )  ^ {-1} | P ( \xi ) |  ^ {2s} \widehat \phi    d \xi .
$$

The constant term  $  \mu _ {0} $
at  $  s = 0 $
gives a [[Fundamental solution|fundamental solution]] to  $  P( D) $. 
Using results about regular holonomic modules it can be proved that the analytic wave front set of  $  \mu _ {0} $
is equal to its  $  C  ^  \infty  $
wave front set when the polynomial is homogeneous. For more results about Fourier transforms and  $  D $-
modules with polynomial coefficients see [[#References|[a32]]] (and also [[D-module| $  D $-
module]]).

The Weyl algebra is a special case of rings of differential operators on non-singular algebraic varieties. See [[#References|[a36]]] and [[#References|[a20]]] for such constructions, which were made before the detailed study of Weyl algebras started.

But foremost  $  A _ {n} ( K) $
is fundamental in algebraic  $  D $-
module theory. The reason is that any quasi-projective manifold  $  X $
over  $  \mathbf C $
is covered in the [[Zariski topology|Zariski topology]] by charts given by affine manifolds  $  V $
for which there exists an unramified covering mapping onto  $  \mathbf C  ^ {n} \setminus  T $, 
where  $  n = d _ {X} $
and  $  T \subset  \mathbf C  ^ {n} $
is an algebraic hypersurface. The ring of differential operators on the affine algebra  $  {\mathcal O} ( V) $
of regular functions on  $  V $
is equal to  $  {\mathcal O} ( V) \otimes _ {\mathbf C [ x] }  A _ {n} ( \mathbf C ) $. 
For algebraic  $  D $-
module theory see [[#References|[a9]]].

==Fuchsian filtrations.==
Above, the filtration on  $  A _ {n} ( K) $
defined by the usual order of differential operators was considered. The Weyl algebra can be endowed with other filtrations, which no longer are positive. With  $  X = \mathbf C  ^ {n+1} $
and coordinates  $  x _ {1} \dots x _ {n} , t $
one takes the hyperplane  $  \{ t = 0 \} $, 
and the filtration on  $  A _ {n+1} ( \mathbf C ) $
is such that  $   \mathop{\rm deg} ( t) = - 1 $, 
$   \mathop{\rm deg} ( \partial  / \partial  t) = 1 $, 
while  $  x _ {v} $
and  $  \partial  / \partial  x _ {v} $
are of degree zero for every  $  v $. 
The associated graded ring is  $  A _ {n+1} ( \mathbf C ) $. 
The associated Rees ring of the Fuchsian filtered ring is Noetherian and its global homological dimension is  $  ( 2n+ 1 ) $. 
This ring has interest in its own and appears in more extensive classes of rings, [[#References|[a41]]]. The Fuchsian filtration is adapted to study nearby and vanishing cycles (cf. [[Vanishing cycle]]). If  $  M $
is a holonomic  $  A _ {n+1} ( \mathbf C ) $-
module, then there exists a unique good filtration  $  V _ {\mathbf . }  ( M) $
with respect to the Fuchsian filtration such that the minimal polynomial of the Euler mapping on  $  \oplus V _ {k} ( M)/V _ {k-1} ( M) $
has roots in the lattice  $  \{ 0 \leq   \mathop{\rm Re} ( \lambda ) < 1 \} $. 
One refers to this as the Kashiwara–Malgrange filtration. Every homogeneous quotient  $  V _ {k} ( M) / V _ {k-1} ( M) $
is a holonomic module over the  $  n $-
dimensional Weyl algebra in the  $  x $-
variables. When  $  M $
is regular holonomic and the Riemann–Hilbert correspondence is applied, one proves that the de Rham complex of  $  V _ {0} ( M)/ V _ {-1} ( M) $
is the nearby cycle along  $  \{ t = 0 \} $
of the perverse sheaf complex defined by the de Rham complex of  $  M $. 
For further details see [[#References|[a16]]].

One should also mention the Bernstein filtration on the Weyl algebra, where both  $  x _ {v} $
and  $  \partial  _ {v} $
have degree  $  + 1 $. 
For  $  n = 1 $, 
[[#References|[a44]]] contains a description of the graded ideals of  $  K[ x, y] $
which are generated by principal symbols of elements in a left ideal of  $  A _ {1} ( K) $, 
using the Bernstein filtration to identify  $  K[ x, y] $
with  $   \mathop{\rm gr} ( A _ {1} ( K)) $.

Weyl algebras with coefficients in a ring exist, i.e. for any ring  $  R $
the ring  $  A _ {n} ( R) = A _ {n} ( \mathbf Z ) \otimes _ {\mathbf Z }  R $
exists. New phenomena may occur when  $  R $
is non-commutative. The work [[#References|[a19]]] shows that if  $  D $
is the skew-field given by the quotient field  $  D _ {1} ( K) $, 
then the ring  $  A _ {1} ( D) $
has global dimension  $  2 $. 
For further results of Weyl algebras over division rings see [[#References|[a18]]]. If  $  R $
is a commutative Noetherian and regular  $  \mathbf Q $-
algebra, one has

$$ 
 \mathop{\rm gl}.dim ( A _ {n} ( R))  =  n +  \mathop{\rm gl}.dim ( R).
$$

See [[#References|[a6]]] and [[#References|[a17]]] for this result and various extensions, where the global homological dimension is computed for other classes of rings of differential operators. If  $  S $
is a non-commutative  $  \mathbf Q $-
algebra equipped with a Zariskian filtration such that  $   \mathop{\rm gr} ( S) $
is a commutative regular Noetherian ring, one constructs  $  A _ {1} ( S) = A _ {1} ( \mathbf Q ) \otimes _ {\mathbf Q }  S $. 
It is an open problem if  $   \mathop{\rm gl}.dim ( A _ {1} ( S))=  \mathop{\rm gl}.dim ( S) + 1 $
holds. The work [[#References|[a13]]] contains results which predict the grade number of graded modules over  $  A _ {1} ( S) $
where the Fuchsian graded structure on  $  A _ {1} ( \mathbf Q ) $
induces a graded ring structure on  $  A _ {1} ( S) $.

Invariant theory is another topic where the Weyl algebra appears. In [[#References|[a29]]] it is proved that if  $  G $
is a finite group of automorphisms on  $  \mathbf C [ x _ {1} \dots x _ {n} ] $
which does not contain any pseudo-reflection different from the identity, then the ring of differential operators on the  $  G $-
invariant subring of  $  \mathbf C [ x] $
is equal to the  $  G $-
invariant subring  $   \mathop{\rm pf}  A _ {n} ( \mathbf C ) $. 
Moreover, the ring  $  A _ {n} ( \mathbf C )  ^ {G} $
is an Auslander–Gorenstein ring whose injective dimension is  $  n $. 
This means that the bimodule given by the ring  $  A _ {n} ( \mathbf C )  ^ {G} $
has an injective resolution of length  $  n $
both as a left and a right module, and Auslander's condition holds:

$$ 
 \mathop{\rm Ext} _ {A}  ^ {v} ( N,  \mathop{\rm Ext} _ {A}  ^ {k} ( M, A))
$$

$$ 
\textrm{ for  all   } v < k  \textrm{ and }  N  \subset    \mathop{\rm Ext} _ {A}  ^ {k} ( M , A) ,
$$

where  $  A = A _ {N} ( \mathbf C )  ^ {G} $
and  $  M $
is any finitely-generated  $  A $-
module. This condition was originally verified for an extensive class of filtered rings, including the Weyl algebra in [[#References|[a39]]]. See [[#References|[a30]]] for more facts about rings of differential operators related to invariant theory.

==Noetherian operators.==
The Weyl algebra is used in commutative algebra to describe primary ideals by equations with respect to its prime radical. Let  $  \mathfrak q \subset  \mathbf C [ x _ {1} \dots x _ {n} ] $
be a [[Primary ideal|primary ideal]] and set  $  \sqrt \mathfrak q = \mathfrak p $. 
Then there exists a finite set  $  Q _ {1} \dots Q _ {s} $
in  $  A _ {n} ( \mathbf C ) $
such that  $  Q _ {v} ( \mathfrak q ) \subset  \mathfrak p $
for every  $  v $. 
Conversely, let  $  P \in \mathbf C [ x] $
be such that  $  Q _ {v} ( P) \in \mathfrak p $
for every  $  v $. 
Then  $  P \in \mathfrak q $. 
So  $  \mathbf C [ x]/ \mathfrak q $
becomes a submodule of a direct sum of  $  \mathbf C [ x]/ \mathfrak p $. 
This fact is used in the fundamental principle by Ehrenpreis to represent solutions to homogeneous systems of partial differential equations by absolutely convergent integrals over exponential solutions. See [[#References|[a23]]], which also includes a construction of Noetherian operators.

==Enveloping algebras.==
If  $  \mathfrak g $
is a Heisenberg algebra, i.e. a nilpotent finite-dimensional Lie algebra with  $  1 $-
dimensional centre  $  \mathfrak c $
such that  $  [ \mathfrak g , \mathfrak g ] = \mathfrak c $, 
then  $  A _ {n} ( K) $
is a quotient ring of the enveloping algebra over  $  \mathfrak g $. 
Hence  $  A _ {n} ( K) $-
modules yield representations of  $  \mathfrak g $, 
which are infinite dimensional when  $  K $
has characteristic zero. See [[#References|[a11]]] for this. Quotients by primitive ideals of enveloping algebras of semi-simple Lie algebras lead to more involved results. The Weyl algebra  $  A _ {1} ( K) $
appears in the study of  $  U( \mathfrak g ) / ( Q- \lambda ) $, 
where  $  \mathfrak g = \mathfrak s \mathfrak l ( 2 , \mathbf C ) $, 
$  Q $
is the Casimir operator and  $  \lambda \in \mathbf C $. 
This ring is a subring of  $  A _ {1} ( \mathbf C ) $. 
See [[#References|[a38]]], [[#References|[a39]]]. Several problems concerned with semi-simple elements of [[#References|[a11]]] were proposed in [[#References|[a12]]]. For affirmative answers in some cases and related problems to Weyl algebras in many variables see [[#References|[a24]]]. Finally one should mention the fundamental result in [[#References|[a2]]], which is crucial for applications of algebraic  $  D $-
module theory to representation theory of Lie algebras. See also [[#References|[a25]]]. This gives special interest to the Weyl algebra and related rings, such as the ring of differential operators on the projective space  $  \mathbf P _ {n} ( \mathbf C ) $, 
which were determined in . See also [[#References|[a21]]] for  $  D $-module theory related to representations of Lie algebras.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Barlet, "
Monodromie et pôles du prolongement méromorphe de $\int_X |f|^{2\lambda}$" ''Bull. Soc. Math. France'' , '''114''' (1986) pp. 247–269 {{ZBL|0652.32010}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> A.A. Beilinson, J. Bernstein, "Localisation des $\mathfrak{g}$-modules" ''C.R. Acad. Sci. Paris'' , '''292''' (1981) pp. 15–18 {{MR|610137}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> I.N. Bernstein, "Modules over a ring of differential operators. Study of the fundamental solutions to equations with constant coefficients" ''Funct. Anal. Appl.'' , '''5''' : 2 (1971) pp. 89–101 ''Funkts. Anal. i Prilozh.'' , '''5''' : 2 (1971) pp. 1–16 {{MR|0290097}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Bernstein, "The analytic continuation of generalized functions with respect to a parameter" ''Funct. Anal. Appl.'' , '''6''' : 4 (1972) pp. 273–285 ''Funkts. Anal. i Prilozh.'' , '''6''' : 4 (1972) pp. 3–25 {{MR|}} {{ZBL|0282.46038}} </TD></TR>
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