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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w0976701.png" /> be a commutative field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w0976702.png" /> a positive integer. The ring of differential operators with coefficients in the polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w0976703.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w0976704.png" /> and called the Weyl algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w0976705.png" /> variables over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w0976706.png" />. Identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w0976707.png" /> with the subring of zero-order differential operators, it follows that the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w0976708.png" /> is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w0976709.png" /> and the derivation operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767010.png" />. The commutators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767011.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767012.png" />. So <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767013.png" /> is a non-commutative ring. Every element has a unique representation
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767014.png" /></td> </tr></table>
+
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767015.png" /> are monomials in the derivation operators. The largest integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767016.png" /> such that a polynomial coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767017.png" /> is non-zero with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767018.png" /> is the order of the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767019.png" />. The order yields a filtration (cf. [[Filtered module|Filtered module]]) and the associated graded ring (cf. [[Graded module|Graded module]])
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767020.png" /></td> </tr></table>
+
$$
 +
P( x, \partial  )  = \sum _ {v = 0 } ^ { m }  p _  \alpha  ( x) \partial  ^  \alpha  ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767021.png" /> is the set of operators of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767022.png" />, divided by those of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767023.png" /> at most. It is well-known that the associated graded ring is isomorphic to the polynomial ring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767024.png" /> variables over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767026.png" /> are the generators.
+
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.==
 
==Ring-theoretic properties.==
Here only the case when the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767027.png" /> has characteristic zero is discussed. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767028.png" />, the results below are no longer valid. For material when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767029.png" /> see [[#References|[a30]]]. From now on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767030.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767031.png" /> is a [[Simple ring|simple ring]] and since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767032.png" /> is Noetherian and commutative, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767033.png" /> is both left and right Noetherian. By [[#References|[a42]]], every left ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767034.png" /> is generated by two elements. The global [[Homological dimension|homological dimension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767035.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767036.png" />. This result was proved in [[#References|[a37]]]. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767037.png" /> was settled before in [[#References|[a35]]]. Another important result is the involutivity of characteristic ideals.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767038.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767039.png" />. A good filtration on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767040.png" /> consists of an increasing sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767041.png" />-submodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767042.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767043.png" /> for all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767044.png" />, and the associated graded module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767045.png" /> is finitely generated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767046.png" />. A module can be equipped with different good filtrations. But there exists a unique graded ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767047.png" />, given as the radical of the annihilating ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767048.png" /> for any good filtration. It is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767049.png" /> and called the characteristic ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767050.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767051.png" /> there exists a Poisson product such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767052.png" />. The involutivity theorem asserts that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\{ J ( M) , J( M) \}  \subset  J( M)
 +
$$
  
for every finitely-generated left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767054.png" />-module. In the special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767055.png" />, the Nullstellen Satz (cf. [[Hilbert theorem|Hilbert theorem]]) identifies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767056.png" /> with an algebraic set in the symplectic cotangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767057.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767058.png" /> and called the characteristic variety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767059.png" /> (cf. also [[Characteristic manifold|Characteristic manifold]]). Then (a1) means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767060.png" /> is involutive in the symplectic cotangent space.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767061.png" /> is at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767062.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767063.png" /> is a non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767064.png" />-module. This can be used to prove that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767065.png" />, using the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767066.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767067.png" /> is a multiplicative set formed by homogeneous elements and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767068.png" /> is the multiplicative set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767069.png" /> whose principal symbols belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767070.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767071.png" /> satisfies the two-sided Ore condition. So the universal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767072.png" />-inverting ring is the two-sided ring of Ore fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767073.png" />. In particular, this applies when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767074.png" /> is the set of non-zero elements. The resulting division ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767075.png" /> is related to division rings of certain enveloping algebras divided by primitive ideals.
+
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.==
 
==Holonomic modules.==
The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767076.png" /> implies that its Krull dimension is at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767077.png" />. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767078.png" /> was actually proved before the discoveries above in [[#References|[a15]]], using a trace formula which shows that every non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767079.png" />-module is an infinite-dimensional vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767080.png" />. Of special interest is the set of finitely-generated left or right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767081.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767082.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767083.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767084.png" /> which are simple. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767085.png" /> is any positive integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767086.png" /> is a principal ideal, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767087.png" /> is non-holonomic when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767088.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767089.png" />, then the subring of rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767090.png" /> is a holonomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767091.png" />-module. This was proved by J. Bernstein in [[#References|[a3]]], [[#References|[a4]]], in which also a functional equation was found expressed by the equality:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767092.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
b( s) P( x)  ^ {s}  = \sum s  ^ {v} Q _ {v} ( x, \partial  ) ( P( x)  ^ {s+1} ) .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767093.png" /> is a polynomial, chosen to have smallest possible degree and highest coefficient one. It is called the Bernstein–Sato polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767094.png" />. The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767095.png" /> is of particular interest. It is proved in [[#References|[a27]]] that the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767096.png" /> consist of strictly negative rational numbers. See also [[#References|[a6]]] for this. The roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767097.png" /> are related to the monodromy acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767098.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w09767099.png" />, where it is assumed that zero is the only critical value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670100.png" />. In [[#References|[a31]]] it is proved that the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670101.png" /> is equal to the union of eigenvalues of monodromy in every dimension in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670102.png" />. Bernstein's functional equation gives meromorphic continuations of distributions. The meromorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670103.png" />-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670104.png" /> exists, with poles contained in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670105.png" />. The roots of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670106.png" />-function give an effective contribution to the set of poles. Namely, for any root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670107.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670108.png" /> there exists some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670109.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670110.png" /> is a pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670111.png" />. This is proved in [[#References|[a1]]].
+
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.==
 
==Fundamental solutions.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670112.png" /> be a differential operator with constant coefficients. Using the Fourier transform, and replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670113.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670114.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670115.png" />-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670116.png" /> exists, acting on test-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670117.png" /> by
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670118.png" /></td> </tr></table>
+
$$
 +
\langle  \mu _ {s} , \phi \rangle  = \int\limits _ {\mathbf R  ^ {n} }
 +
P( \xi )  ^ {-1} | P ( \xi ) |  ^ {2s} \widehat \phi    d \xi .
 +
$$
  
The constant term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670119.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670120.png" /> gives a [[Fundamental solution|fundamental solution]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670121.png" />. Using results about regular holonomic modules it can be proved that the analytic wave front set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670122.png" /> is equal to its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670123.png" /> wave front set when the polynomial is homogeneous. For more results about Fourier transforms and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670124.png" />-modules with polynomial coefficients see [[#References|[a32]]] (and also [[D-module|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670125.png" />-module]]).
+
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670126.png" /> is fundamental in algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670127.png" />-module theory. The reason is that any quasi-projective manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670128.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670129.png" /> is covered in the [[Zariski topology|Zariski topology]] by charts given by affine manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670130.png" /> for which there exists an unramified covering mapping onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670131.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670132.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670133.png" /> is an algebraic hypersurface. The ring of differential operators on the affine algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670134.png" /> of regular functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670135.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670136.png" />. For algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670137.png" />-module theory see [[#References|[a9]]].
+
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.==
 
==Fuchsian filtrations.==
Above, the filtration on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670138.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670139.png" /> and coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670140.png" /> one takes the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670141.png" />, and the filtration on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670142.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670144.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670145.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670146.png" /> are of degree zero for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670147.png" />. The associated graded ring is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670148.png" />. The associated Rees ring of the Fuchsian filtered ring is Noetherian and its global homological dimension is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670149.png" />. 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|Vanishing cycle]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670150.png" /> is a holonomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670151.png" />-module, then there exists a unique good filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670152.png" /> with respect to the Fuchsian filtration such that the minimal polynomial of the Euler mapping on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670153.png" /> has roots in the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670154.png" />. One refers to this as the Kashiwara–Malgrange filtration. Every homogeneous quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670155.png" /> is a holonomic module over the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670156.png" />-dimensional Weyl algebra in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670157.png" />-variables. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670158.png" /> is regular holonomic and the Riemann–Hilbert correspondence is applied, one proves that the de Rham complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670159.png" /> is the nearby cycle along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670160.png" /> of the perverse sheaf complex defined by the de Rham complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670161.png" />. For further details see [[#References|[a16]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670162.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670163.png" /> have degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670164.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670165.png" />, [[#References|[a44]]] contains a description of the graded ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670166.png" /> which are generated by principal symbols of elements in a left ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670167.png" />, using the Bernstein filtration to identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670168.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670169.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670170.png" /> the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670171.png" /> exists. New phenomena may occur when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670172.png" /> is non-commutative. The work [[#References|[a19]]] shows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670173.png" /> is the skew-field given by the quotient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670174.png" />, then the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670175.png" /> has global dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670176.png" />. For further results of Weyl algebras over division rings see [[#References|[a18]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670177.png" /> is a commutative Noetherian and regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670178.png" />-algebra, one has
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670179.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm gl}.dim ( A _ {n} ( R))  = n +  \mathop{\rm gl}.dim ( R).
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670180.png" /> is a non-commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670181.png" />-algebra equipped with a Zariskian filtration such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670182.png" /> is a commutative regular Noetherian ring, one constructs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670183.png" />. It is an open problem if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670184.png" /> holds. The work [[#References|[a13]]] contains results which predict the grade number of graded modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670185.png" /> where the Fuchsian graded structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670186.png" /> induces a graded ring structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670187.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670188.png" /> is a finite group of automorphisms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670189.png" /> which does not contain any pseudo-reflection different from the identity, then the ring of differential operators on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670190.png" />-invariant subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670191.png" /> is equal to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670192.png" />-invariant subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670193.png" />. Moreover, the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670194.png" /> is an Auslander–Gorenstein ring whose injective dimension is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670195.png" />. This means that the bimodule given by the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670196.png" /> has an injective resolution of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670197.png" /> both as a left and a right module, and Auslander's condition holds:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670198.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ext} _ {A}  ^ {v} ( N,  \mathop{\rm Ext} _ {A}  ^ {k} ( M, A))
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670199.png" /></td> </tr></table>
+
$$
 +
\textrm{ for  all  } v < k  \textrm{ and }  N  \subset    \mathop{\rm Ext} _ {A}  ^ {k} ( M , A) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670200.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670201.png" /> is any finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670202.png" />-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.
+
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.==
 
==Noetherian operators.==
The Weyl algebra is used in commutative algebra to describe primary ideals by equations with respect to its prime radical. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670203.png" /> be a [[Primary ideal|primary ideal]] and set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670204.png" />. Then there exists a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670205.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670206.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670207.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670208.png" />. Conversely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670209.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670210.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670211.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670212.png" />. So <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670213.png" /> becomes a submodule of a direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670214.png" />. 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.
+
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.==
 
==Enveloping algebras.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670215.png" /> is a Heisenberg algebra, i.e. a nilpotent finite-dimensional Lie algebra with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670216.png" />-dimensional centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670217.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670218.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670219.png" /> is a quotient ring of the enveloping algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670220.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670221.png" />-modules yield representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670222.png" />, which are infinite dimensional when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670223.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670224.png" /> appears in the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670225.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670226.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670227.png" /> is the Casimir operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670228.png" />. This ring is a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670229.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670230.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670231.png" />, which were determined in . See also [[#References|[a21]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670232.png" />-module theory related to representations of Lie 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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Barlet,   "Monodromic et pôles de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670233.png" />"  ''Bull. Soc. Math. France'' , '''114''' (1986) pp. 247–269</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.A. Beilinson,   J. Bernstein,   "Localisation des <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670234.png" />-modules" ''C.R. Acad. Sci. Paris'' , '''292''' (1981) pp. 15–18</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</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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> F. Bien,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670235.png" />-modules and spherical representations of symmetric spaces" , Princeton Univ. Press (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J.-E. Björk,   "Rings of differential operators" , North-Holland (1979)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> J.-E. Björk,   "The global homological dimension of some algebras of differential operators" ''Invent. Math.'' , '''17''' (1972) pp. 67–78</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J.-E. Björk,   "Non-commutative Noetherian rings and their use in homological algebra" ''J. Pure Appl. Algebra'' , '''38''' (1985) pp. 111–119</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> A. Borel,   et al.,   "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670236.png" />-modules" , Acad. Press (1987)</TD></TR><TR><TD valign="top">[a10a]</TD> <TD valign="top"> W. Borho,   J.-L Brylinski,   "Differential operators on homogeneous spaces I" ''Invent. Math.'' , '''69''' (1982) pp. 437–476</TD></TR><TR><TD valign="top">[a10b]</TD> <TD valign="top"> W. Borho,   J.-L Brylinski,   "Differential operators on homogeneous spaces II" ''Invent. Math.'' , '''80''' (1985) pp. 1–68</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> J. Dixmier,   "Algèbres enveloppantes" , Gauthier-Villars (1974)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> J. Dixmier,   "Sur les algèbres de Weyl II" ''Bull. Sci. Math.'' , '''94''' (1970) pp. 289–301</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> E.K. Ekström,   "Homological properties of some Weyl algebra extensions" ''Compositio Math.'' , '''75''' (1989) pp. 231–246</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> O. Gabber,   "The integrability of the characteristic variety" ''Amer. J. Math.'' , '''103''' (1981) pp. 445–468</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P. Gabriel,   R. Rentschler,   "Sur la dimension des anneaux et ensembles ordonnes" ''C.R. Acad. Sci. Paris'' , '''265''' (1967) pp. A712-A715</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> V. Ginsburg,   "Characteristic varieties and vanishing cycles" ''Inv. Math.'' , '''84''' (1986) pp. 327–403</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> K.R. Goodearl,   R.B. Warfield Jr.,   "Krull dimension of differential operator rings" ''Proc. London Math. Soc.'' , '''45''' (1982) pp. 49–70</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> K.R. Goodearl,   T.J. Hodges,   T.H. Lenagan,   "Krull and global dimensions of Weyl algebras over division rings" ''J. Algebra'' , '''91''' (1984) pp. 334–359</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> R. Hart,   "A note on tensor products of algebras" ''J. Algebra'' , '''21''' (1972) pp. 422–427</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> G. Hochschild,   B. Kostant,   B. Rosenber,   "Differential forms on regular affine algebras" ''Trans. Amer. Math. Soc.'' , '''102''' (1962) pp. 383–408</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> R. Hotta,   M. Kashiwara,   "The invariant system on a semi-simple Lie algebra" ''Inv. Math.'' , '''75''' (1984) pp. 327–358</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer (1983)</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> L. Hörmander,   "An introduction to complex analysis in several variables" , North-Holland (1990)</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> A. Joseph,   "The Weyl algebra - semisimple and nilpotent elements" ''Amer. J. Math.'' , '''97''' (1975) pp. 597–615</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> A. Joseph,   "Primitive ideals in enveloping algebras" , ''Proc. Internat. Congress Mathematicians (Warszawa, 1983)'' , '''1''' , PWN &amp; North-Holland (1984) pp. 403–414</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> M. Kashiwara,   "A study of over-determined systems" , Kyoto University (1970) (Thesis)</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> M. Kashiwara,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670237.png" />-functions and holonomic systems" ''Inv. Math.'' , '''38''' (1975) pp. 121–135</TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top"> M. Kashiwara,   "Regular holonomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670238.png" />-modules and distributions on complex manifolds" T. Suwa (ed.) P. Wagreich (ed.) , ''Complex analytic singularities'' , ''Adv. Studies in Math.'' , '''8''' , Kinokuniya &amp; North-Holland (1987) pp. 199–206</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> T. Levasseur,   "Anneaux d'operateurs differentiels" M.P. Malliavin (ed.) , ''Sem. P. Dubreil et M.P. Malliavin'' , ''Lect. notes in math.'' , '''867''' , Springer (157–173)</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> T. Levasseur,   J.T. Stafford,   "Rings of differential operators on classical rings of invariants" ''Memoirs Amer. Math. Soc.'' , '''41''' (1989)</TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top"> B. Malgrange,   "Polynome de Bernstein–Sato et cohomologie evanescente" ''Asterisque'' , '''101–102''' (1983) pp. 243–267</TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top"> B. Malgrange,   "Equations différentiels à coefficients polynomiaux" , Birkhäuser (1991)</TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top"> J.C. McConnel,   "Noncommutative Noetherian rings" , Wiley (1987)</TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top"> P. Revoy,   "Algèbres de Weyl en caracteristique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670239.png" />" ''C.R. Acad. Sci. Paris'' , '''276''' (1973) pp. 225–228</TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top"> G.S. Rinehart,   "Note on the global dimension of a certain ring" ''Proc. Amer. Math. Soc.'' , '''13''' (1963) pp. 195–222</TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top"> G.S. Rinehart,   "Differential forms on general commutative algebras" ''Trans. Amer. Math. Soc.'' , '''103''' (1963) pp. 195–222</TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top"> J.-E. Roos,   "Determination de la dimension homologique globale des algèbres de Weyl" ''C.R. Acad. Sci. Paris'' , '''274''' (1972) pp. A23-A26</TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top"> J.-E. Roos,   "Properties homologiques des quotients primitifs des algèbres enveloppantes des algèbres de Lie semi-simples" ''C.R. Acad. Sci. Paris'' , '''276''' (1973) pp. 351–354</TD></TR><TR><TD valign="top">[a39]</TD> <TD valign="top"> J.-E. Roos,   "Complements" ''C.R. Acad. Sci. Paris'' , '''276''' (1973) pp. 447–450</TD></TR><TR><TD valign="top">[a40]</TD> <TD valign="top"> M. Kashiwara,   "Microfunctions and pseudo-differential equations" H. Komatsu (ed.) , ''Hyperfunctions and pseudodifferential equations. Proc. Conf. Katata, 1971'' , ''Lect. notes in math.'' , '''287''' , Springer (1973) pp. 265–529</TD></TR><TR><TD valign="top">[a41]</TD> <TD valign="top"> S.P. Smith,   "Differential operators in commutative algebras" , ''Lect. notes in math.'' , '''1197''' , Springer (1986) pp. 165–177</TD></TR><TR><TD valign="top">[a42]</TD> <TD valign="top"> J.L. Stafford,   "Non-holonomic modules over Weyl algebras and enveloping algebras" ''Inv. Math.'' , '''79''' (1985) pp. 619–638</TD></TR><TR><TD valign="top">[a43]</TD> <TD valign="top"> J.L. Stafford,   "Module structure over Weyl algebras" ''Leeds Univ. preprint'' (1977)</TD></TR><TR><TD valign="top">[a44]</TD> <TD valign="top"> P. Strömbeck,   "On left ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670240.png" /> and their associated graded ideals" ''J. Algebra'' , '''55''' (1978) pp. 116–144</TD></TR><TR><TD valign="top">[a45]</TD> <TD valign="top"> J.-E. Björk,   "Analytic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670241.png" />-modules" , Kluwer (1993)</TD></TR></table>
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 +
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>
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<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>
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<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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<TR><TD valign="top">[a5]</TD> <TD valign="top"> F. Bien, "$D$-modules and spherical representations of symmetric spaces" , Princeton Univ. Press (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J.-E. Björk, "Rings of differential operators" , North-Holland (1979) {{MR|0549189}} {{ZBL|0499.13009}} </TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top"> J.-E. Björk, "The global homological dimension of some algebras of differential operators" ''Invent. Math.'' , '''17''' (1972) pp. 67–78 {{MR|0453846}} {{MR|0320078}} {{ZBL|0255.13010}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J.-E. Björk, "Non-commutative Noetherian rings and their use in homological algebra" ''J. Pure Appl. Algebra'' , '''38''' (1985) pp. 111–119</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> A. Borel, et al., "Algebraic $D$-modules" , Acad. Press (1987) {{MR|882000}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a10a]</TD> <TD valign="top"> W. Borho, J.-L Brylinski, "Differential operators on homogeneous spaces I" ''Invent. Math.'' , '''69''' (1982) pp. 437–476 {{MR|0679767}} {{ZBL|0504.22015}} </TD></TR><TR><TD valign="top">[a10b]</TD> <TD valign="top"> W. Borho, J.-L Brylinski, "Differential operators on homogeneous spaces II" ''Invent. Math.'' , '''80''' (1985) pp. 1–68 {{MR|1015807}} {{ZBL|0702.22019}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> J. Dixmier, "Algèbres enveloppantes" , Gauthier-Villars (1974) {{MR|0498737}} {{MR|0498740}} {{MR|0498742}} {{ZBL|0308.17007}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> J. Dixmier, "Sur les algèbres de Weyl II" ''Bull. Sci. Math.'' , '''94''' (1970) pp. 289–301 {{MR|0299632}} {{ZBL|0202.04303}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> E.K. Ekström, "Homological properties of some Weyl algebra extensions" ''Compositio Math.'' , '''75''' (1989) pp. 231–246 {{MR|1065208}} {{ZBL|0717.16032}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> O. Gabber, "The integrability of the characteristic variety" ''Amer. J. Math.'' , '''103''' (1981) pp. 445–468 {{MR|0618321}} {{ZBL|0492.16002}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P. Gabriel, R. 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Algebra'' , '''21''' (1972) pp. 422–427</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> G. Hochschild, B. Kostant, B. Rosenber, "Differential forms on regular affine algebras" ''Trans. Amer. Math. Soc.'' , '''102''' (1962) pp. 383–408 {{MR|0142598}} {{ZBL|0102.27701}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> R. Hotta, M. Kashiwara, "The invariant system on a semi-simple Lie algebra" ''Inv. Math.'' , '''75''' (1984) pp. 327–358</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> L.V. Hörmander, "The analysis of linear partial differential operators" , '''1''' , Springer (1983) {{MR|0717035}} {{MR|0705278}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1990) {{MR|1045639}} {{ZBL|0685.32001}} </TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> A. Joseph, "The Weyl algebra - semisimple and nilpotent elements" ''Amer. J. Math.'' , '''97''' (1975) pp. 597–615 {{MR|0379615}} {{ZBL|0316.16036}} </TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> A. Joseph, "Primitive ideals in enveloping algebras" , ''Proc. Internat. Congress Mathematicians (Warszawa, 1983)'' , '''1''' , PWN &amp; North-Holland (1984) pp. 403–414 {{MR|0804696}} {{ZBL|0574.17007}} </TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> M. Kashiwara, "A study of over-determined systems" , Kyoto University (1970) (Thesis)</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> M. Kashiwara, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670237.png" />-functions and holonomic systems" ''Inv. Math.'' , '''38''' (1975) pp. 121–135 {{MR|0430304}} {{ZBL|0354.35082}} </TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top"> M. Kashiwara, "Regular holonomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670238.png" />-modules and distributions on complex manifolds" T. Suwa (ed.) P. Wagreich (ed.) , ''Complex analytic singularities'' , ''Adv. Studies in Math.'' , '''8''' , Kinokuniya &amp; North-Holland (1987) pp. 199–206 {{MR|894293}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> T. Levasseur, "Anneaux d'operateurs differentiels" M.P. Malliavin (ed.) , ''Sem. P. Dubreil et M.P. Malliavin'' , ''Lect. notes in math.'' , '''867''' , Springer (157–173) {{MR|0910403}} {{MR|0755924}} {{MR|0633520}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> T. Levasseur, J.T. Stafford, "Rings of differential operators on classical rings of invariants" ''Memoirs Amer. Math. Soc.'' , '''41''' (1989) {{MR|0988083}} {{ZBL|0691.16019}} </TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top"> B. Malgrange, "Polynome de Bernstein–Sato et cohomologie evanescente" ''Asterisque'' , '''101–102''' (1983) pp. 243–267 {{MR|0737934}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top"> B. Malgrange, "Equations différentiels à coefficients polynomiaux" , Birkhäuser (1991) {{MR|1117227}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top"> J.C. McConnel, "Noncommutative Noetherian rings" , Wiley (1987)</TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top"> P. Revoy, "Algèbres de Weyl en caracteristique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670239.png" />" ''C.R. Acad. Sci. Paris'' , '''276''' (1973) pp. 225–228 {{MR|0335564}} {{ZBL|0265.16007}} </TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top"> G.S. Rinehart, "Note on the global dimension of a certain ring" ''Proc. Amer. Math. Soc.'' , '''13''' (1963) pp. 195–222 {{MR|0137747}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top"> G.S. Rinehart, "Differential forms on general commutative algebras" ''Trans. Amer. Math. Soc.'' , '''103''' (1963) pp. 195–222 {{MR|0154906}} {{ZBL|0113.26204}} </TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top"> J.-E. Roos, "Determination de la dimension homologique globale des algèbres de Weyl" ''C.R. Acad. Sci. Paris'' , '''274''' (1972) pp. A23-A26 {{MR|0292914}} {{ZBL|0227.16021}} </TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top"> J.-E. Roos, "Properties homologiques des quotients primitifs des algèbres enveloppantes des algèbres de Lie semi-simples" ''C.R. Acad. Sci. Paris'' , '''276''' (1973) pp. 351–354</TD></TR><TR><TD valign="top">[a39]</TD> <TD valign="top"> J.-E. Roos, "Complements" ''C.R. Acad. Sci. Paris'' , '''276''' (1973) pp. 447–450 {{MR|0318250}} {{ZBL|0261.17011}} </TD></TR><TR><TD valign="top">[a40]</TD> <TD valign="top"> M. Kashiwara, "Microfunctions and pseudo-differential equations" H. Komatsu (ed.) , ''Hyperfunctions and pseudodifferential equations. Proc. Conf. Katata, 1971'' , ''Lect. notes in math.'' , '''287''' , Springer (1973) pp. 265–529 {{MR|0420735}} {{ZBL|0277.46039}} </TD></TR><TR><TD valign="top">[a41]</TD> <TD valign="top"> S.P. Smith, "Differential operators in commutative algebras" , ''Lect. notes in math.'' , '''1197''' , Springer (1986) pp. 165–177</TD></TR><TR><TD valign="top">[a42]</TD> <TD valign="top"> J.T. Stafford, "Non-holonomic modules over Weyl algebras and enveloping algebras" ''Inv. Math.'' , '''79''' (1985) pp. 619–638 {{MR|782240}} {{ZBL|0558.17011}} </TD></TR><TR><TD valign="top">[a43]</TD> <TD valign="top"> J.T. Stafford, "Module structure of Weyl algebras", ''London Math. Soc.'', ''Ser. II'', '''18''' (1978), pp. 429–442 {{MR|0518227}}
 +
</TD></TR><TR><TD valign="top">[a44]</TD> <TD valign="top"> P. Strömbeck, "On left ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670240.png" /> and their associated graded ideals" ''J. Algebra'' , '''55''' (1978) pp. 116–144 {{MR|0515764}} {{ZBL|0399.16002}} </TD></TR><TR><TD valign="top">[a45]</TD> <TD valign="top"> J.-E. Björk, "Analytic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097670/w097670241.png" />-modules" , Kluwer (1993) {{MR|1232191}} {{ZBL|}} </TD></TR></table>

Latest revision as of 19:30, 19 January 2024


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) and the associated graded ring (cf. 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 [a30]. From now on $ \mathop{\rm char} ( K) = 0 $. Then $ A _ {n} ( K) $ is a 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 [a43], every left ideal of $ A _ {n} ( K) $ is generated by two elements. The global homological dimension of $ A _ {n} ( K) $ is equal to $ n $. This result was proved in [a37]. The case $ n = 1 $ was settled before in [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) 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). 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 [a8] for a survey of this. The result (a1) was proved by micro-local analysis in [a40]. An algebraic proof was found later in [a14]. In [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 [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 [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 [a3], [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 [a27] that the roots of $ b( s) $ consist of strictly negative rational numbers. See also [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 [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 [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 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 [a32] (and also $ D $- module).

The Weyl algebra is a special case of rings of differential operators on non-singular algebraic varieties. See [a36] and [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 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 [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, [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 [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 $, [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 [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 [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 [a6] and [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 [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 [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 [a39]. See [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 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 [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 [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 [a38], [a39]. Several problems concerned with semi-simple elements of [a11] were proposed in [a12]. For affirmative answers in some cases and related problems to Weyl algebras in many variables see [a24]. Finally one should mention the fundamental result in [a2], which is crucial for applications of algebraic $ D $- module theory to representation theory of Lie algebras. See also [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 [a21] for $ D $-module theory related to representations of Lie algebras.

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How to Cite This Entry:
Weyl algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_algebra&oldid=16166
This article was adapted from an original article by J.-E. Björk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article