# Weyl algebra

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

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

where $\partial ^ \alpha$ are monomials in the derivation operators. The largest integer $m$ such that a polynomial coefficient $p _ \alpha ( x)$ is non-zero with $| \alpha | = m$ is the order of the differential operator $P$. The order yields a filtration (cf. Filtered module) 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 0$, the results below are no longer valid.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=55213
This article was adapted from an original article by J.-E. BjÃ¶rk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article