Weyl algebra

 ( Algebras ) 

In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics.

In the simplest case, these are differential operators. Let $F$ be a field, and let $Fx$ be the polynomial ring in one variable with coefficients in $F$. Then the corresponding Weyl algebra consists of differential operators of form

$f\_m(x)\; \backslash partial\_x^m\; +\; f\_\{m-1\}(x)\; \backslash partial\_x^\{m-1\}\; +\; \backslash cdots\; +\; f\_1(x)\; \backslash partial\_x\; +\; f\_0(x)$

where $f\_i(x)\backslash in\; Fx$.

This is the first Weyl algebra $A\_1$. The n-th Weyl algebra $A\_n$ is constructed similarly.

Alternatively, $A\_1$ can be constructed as the quotient ring of the free algebra on two generators, q and p, by the ideal generated by $(p,q\; -\; 1)$. Similarly, $A\_n$ is obtained by quotienting the free algebra on 2n generators by the ideal generated by$$(p\_i,q\_j\; -\; \backslash delta\_\{i,j\}),\; \backslash quad\; \backslash forall\; i,\; j\; =\; 1,\; \backslash dots,\; n$$where $\backslash delta\_\{i,j\}$ is the Kronecker delta.

More generally, let $(R,\backslash Delta)$ be a partial differential ring with commuting derivatives $\backslash Delta\; =\; \backslash lbrace\; \backslash partial\_1,\backslash ldots,\backslash partial\_m\; \backslash rbrace$. The Weyl algebra associated to $(R,\backslash Delta)$ is the noncommutative ring $R\backslash partial\_1,\backslash ldots,\backslash partial\_m$ satisfying the relations $\backslash partial\_i\; r\; =\; r\backslash partial\_i\; +\; \backslash partial\_i(r)$ for all $r\; \backslash in\; R$. The previous case is the special case where $R=Fx\_1,\backslash ldots,x\_n$ and $\backslash Delta\; =\; \backslash lbrace\; \backslash partial\_\{x\_1\},\backslash ldots,\backslash partial\_\{x\_n\}\; \backslash rbrace$ where $F$ is a field.

This article discusses only the case of $A\_n$ with underlying field $F$ characteristic zero, unless otherwise stated.

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

---

Motivation The Weyl algebra arises naturally in the context of quantum mechanics and the process of canonical quantization. Consider a classical phase space with canonical coordinates $(q\_1,\; p\_1,\; \backslash dots,\; q\_n,\; p\_n)$. These coordinates satisfy the Poisson bracket relations:$$\backslash \{q\_i,\; q\_j\backslash \}\; =\; 0,\; \backslash quad\; \backslash \{p\_i,\; p\_j\backslash \}\; =\; 0,\; \backslash quad\; \backslash \{q\_i,\; p\_j\backslash \}\; =\; \backslash delta\_\{ij\}.$$In canonical quantization, one seeks to construct a Hilbert space of states and represent the classical observables (functions on phase space) as self-adjoint operators on this space. The canonical commutation relations are imposed:$$\backslash hat\{q\}\_i,\; =\; 0,\; \backslash quad\; \backslash hat\{p\}\_i,\; =\; 0,\; \backslash quad\; \backslash hat\{q\}\_i,\; =\; i\backslash hbar\; \backslash delta\_\{ij\},$$where $\backslash cdot,$ denotes the commutator. Here, $\backslash hat\{q\}\_i$ and $\backslash hat\{p\}\_i$ are the operators corresponding to $q\_i$ and $p\_i$ respectively. Erwin Schrödinger proposed in 1926 the following:

• $\backslash hat\{q\_j\}$ with multiplication by $x\_j$.
• $\backslash hat\{p\}\_j$ with $-i\backslash hbar\; \backslash partial\_\{x\_j\}$.

With this identification, the canonical commutation relation holds.

---

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

---

Representation The Weyl algebra $A\_n$ can be concretely constructed as a representation.

In the differential operator representation, similar to Schrödinger's canonical quantization, let $q\_j$ be represented by multiplication on the left by $x\_j$, and let $p\_j$ be represented by differentiation on the left by $\backslash partial\_\{x\_j\}$.

In the matrix representation, similar to the matrix mechanics, $A\_1$ is represented by

---

Generator $A\_n$ can be constructed as a quotient of a free algebra in terms of generators and relations. One construction starts with an abstract vector space V (of dimension 2 n) equipped with a symplectic form ω. Define the Weyl algebra W( V) to be

$W(V)\; :=\; T(V)\; /\; (\backslash !(\; v\; \backslash otimes\; u\; -\; u\; \backslash otimes\; v\; -\; \backslash omega(v,u),\; \backslash text\{\; for\; \}\; v,u\; \backslash in\; V\; )\backslash !),$

where T( V) is the tensor algebra on V, and the notation $(\backslash !(\; )\backslash !)$ means "the ideal generated by".

In other words, W( V) is the algebra generated by V subject only to the relation . Then, W( V) is isomorphic to A_n via the choice of a Darboux basis for .

$A\_n$ is also a quotient ring of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the central element of the Heisenberg algebra (namely q,) equal to the unit of the universal enveloping algebra (called 1 above).

---

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

The isomorphism is given by the symmetrization map from Sym( V) to W( V)

$a\_1\; \backslash cdots\; a\_n\; \backslash mapsto\; \backslash frac\{1\}\{n!\}\; \backslash sum\_\{\backslash sigma\; \backslash in\; S\_n\}\; a\_\{\backslash sigma(1)\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; a\_\{\backslash sigma(n)\}~.$

If one prefers to have the iħ and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by q _i and iħ∂_qi (as per quantum mechanics usage).

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

Stated in another way, let the Moyal product be denoted $f\; \backslash star\; g$, then the Weyl algebra is isomorphic to $(\backslash mathbb\; Cx\_1,,\; \backslash star)$.

In the case of , the analogous quantization to the Weyl one is the Clifford algebra, which is also referred to as the orthogonal Clifford algebra.

The Weyl algebra is also referred to as the symplectic Clifford algebra. Weyl algebras represent for symplectic the same structure that represent for non-degenerate symmetric bilinear forms.

---

D-module The Weyl algebra can be constructed as a D-module. Specifically, the Weyl algebra corresponding to the polynomial ring $Rx\_1,$ with its usual partial differential structure is precisely equal to Grothendieck's ring of differential operations $D\_\{\backslash mathbb\{A\}^n\_R\; /\; R\}$.

More generally, let $X$ be a smooth scheme over a ring $R$. Locally, $X\; \backslash to\; R$ factors as an étale cover over some $\backslash mathbb\{A\}^n\_R$ equipped with the standard projection. Because " étale" means "(flat and) possessing null cotangent sheaf", this means that every D-module over such a scheme can be thought of locally as a module over the $n^\backslash text\{th\}$ Weyl algebra.

Let $R$ be a commutative algebra over a subring $S$. The ring of differential operators $D\_\{R/S\}$ (notated $D\_R$ when $S$ is clear from context) is inductively defined as a graded subalgebra of $\backslash operatorname\{End\}\_\{S\}(R)$:

• $D^0\_R=R$
• $$

D^k_R=\left\{d \in \operatorname{End}_{S}(R):d, \in D^{k-1}_R \text { for all } a \in R\right\} .

Let $D\_R$ be the union of all $D^k\_R$ for $k\; \backslash geq\; 0$. This is a subalgebra of $\backslash operatorname\{End\}\_\{S\}(R)$.

In the case $R\; =\; Sx\_1,$, the ring of differential operators of order $\backslash leq\; n$ presents similarly as in the special case $S\; =\; \backslash mathbb\{C\}$ but for the added consideration of "divided power operators"; these are operators corresponding to those in the complex case which stabilize $\backslash mathbb\{Z\}x\_1,$, but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit $D\_\{\backslash mathbb\{A\}^n\_\backslash mathbb\{Z\}\; /\; \backslash mathbb\{Z\}\}$. One such example is the operator $\backslash partial\_\{x\_1\}^\{p\}\; :\; x\_1^N\; \backslash mapsto\; \{N\; \backslash choose\; p\}\; x\_1^\{N-p\}$.

Explicitly, a presentation is given by

$D\_\{Sx\_1,/S\}^n\; =\; S\; \backslash langle\; x\_1,\; \backslash dots,\; x\_\backslash ell,\; \backslash \{\backslash partial\_\{x\_i\},\; \backslash partial\_\{x\_i\}^\{2\},\; \backslash dots,\; \backslash partial\_\{x\_i\}^\{n\}\backslash \}\_\{1\; \backslash leq\; i\; \backslash leq\; \backslash ell\}\; \backslash rangle$

with the relations

$x\_i,\; =\; \backslash partial\_\{x\_i\}^\{[k\},\; \backslash partial\_\{x\_j\}^\{m\}]\; =\; 0$

$\backslash partial\_\{x\_i\}^\{k\}\; \backslash partial\_\{x\_i\}^\{m\}\; =\; \{k+m\; \backslash choose\; k\}\; \backslash partial\_\{x\_i\}^\{k+m\}\; ~~~~~\backslash text\{when\; \}k+m\; \backslash leq\; n$

where $\backslash partial\_\{x\_i\}^\{0\}\; =\; 1$ by convention. The Weyl algebra then consists of the limit of these algebras as $n\; \backslash to\; \backslash infty$.

When $S$ is a field of characteristic 0, then $D^1\_R$ is generated, as an $R$-module, by 1 and the $S$-derivations of $R$. Moreover, $D\_R$ is generated as a ring by the $R$-subalgebra $D^1\_R$. In particular, if $S\; =\; \backslash mathbb\{C\}$ and $R=\backslash mathbb\{C\}x\_1,$, then $D^1\_R=R+\; \backslash sum\_i\; R\; \backslash partial\_\{x\_i\}$. As mentioned, $A\_n\; =\; D\_R$.

---

Properties of A_n Many properties of $A\_1$ apply to $A\_n$ with essentially similar proofs, since the different dimensions commute.

---

General Leibniz rule In particular, $q,\; =\; -nq^mp^\{n-1\}$ and $p,\; =\; mq^\{m-1\}p^n$.

---

Degree This allows $A\_1$ to be a graded algebra, where the degree of $\backslash sum\_\{m,\; n\}\; c\_\{m,n\}\; q^m\; p^n$ is $\backslash max\; (m\; +\; n)$ among its nonzero monomials. The degree is similarly defined for $A\_n$.

That is, it has no two-sided nontrivial ideals and has no .

---

Derivation That is, any derivation $D$ is equal to $\backslash cdot,$ for some $f\; \backslash in\; A\_n$; any $f\backslash in\; A\_n$ yields a derivation $\backslash cdot,$; if $f,\; f\text{'}\; \backslash in\; A\_n$ satisfies $\backslash cdot,\; =\; \backslash cdot,$, then $f\; -\; f\text{'}\; \backslash in\; F$.

The proof is similar to computing the potential function for a conservative polynomial vector field on the plane.

Since the commutator is a derivation in both of its entries, $\backslash cdot,$ is a derivation for any $f\backslash in\; A\_n$. Uniqueness up to additive scalar is because the center of $A\_n$ is the ring of scalars.

It remains to prove that any derivation is an inner derivation by induction on $n$.

Base case: Let $D:\; A\_1\; \backslash to\; A\_1$ be a linear map that is a derivation. We construct an element $r$ such that $p,\; =\; D(p),\; q,r\; =\; D(q)$. Since both $D$ and $\backslash cdot,$ are derivations, these two relations generate $g,\; =\; D(g)$ for all $g\backslash in\; A\_1$.

Since $p,\; =\; mq^\{m-1\}p^n$, there exists an element $f\; =\; \backslash sum\_\{m,n\}\; c\_\{m,n\}\; q^m\; p^n$ such that $$$$

     [p, f] =  \sum_{m,n} m c_{m,n} q^m p^n = D(p)
     
     

$$$$

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

Thus, $D(q)\; =\; g(p)\; +\; q,$ for some polynomial $g$. Now, since $q,\; =\; -nq^mp^\{n-1\}$, there exists some polynomial $h(p)$ such that $q,\; =\; g(p)$. Since $p,\; =\; 0$, $r\; =\; f\; +\; h(p)$ is the desired element.

For the induction step, similarly to the above calculation, there exists some element $r\; \backslash in\; A\_n$ such that $q\_1,\; =\; D(q\_1),\; p\_1,\; =\; D(p\_1)$.

Similar to the above calculation, $$$$

[x, D(y) - [y, r]] = 0
      for all $x\; \backslash in\; \backslash \{p\_1,\; q\_1\backslash \},\; y\; \backslash in\; \backslash \{p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash \}$. Since $[x,\; D(y)\; -\; [y,\; r]]$ is a derivation in both $x$ and $y$, $[x,\; D(y)\; -\; [y,\; r]]\; =\; 0$ for all $x\backslash in\; \backslash langle\; p\_1,\; q\_1\backslash rangle$ and all $y\; \backslash in\; \backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$. Here, $\backslash langle\; \backslash rangle$ means the subalgebra generated by the elements.
     

Thus, $\backslash forall\; y\; \backslash in\; \backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$, $$$$

D(y) - [y, r] \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle
     
     

Since $D\; -\; \backslash cdot,$ is also a derivation, by induction, there exists $r\text{'}\; \backslash in\; \backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$ such that $D(y)\; -\; y,\; =\; y,$ for all $y\; \backslash in\; \backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$.

Since $p\_1,\; q\_1$ commutes with $\backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$, we have $D(y)\; =\; y,$ for all $y\; \backslash in\; \backslash \{p\_1,\; \backslash dots,\; p\_n,\; q\_1,\; \backslash dots,\; q\_n\backslash \}$, and so for all of $A\_n$.

---

Representation theory

---

Zero characteristic In the case that the ground field has characteristic zero, the nth Weyl algebra is a simple ring Noetherian ring domain. It has global dimension n, in contrast to the ring it deforms, Sym( V), which has global dimension 2 n.

It has no finite-dimensional representations. Although this follows from simplicity, it can be more directly shown by taking the trace of σ( q) and σ( Y) for some finite-dimensional representation σ (where ).

$\backslash mathrm\{tr\}(\backslash sigma(q),\backslash sigma(Y))=\backslash mathrm\{tr\}(1)~.$

Since the trace of a commutator is zero, and the trace of the identity is the dimension of the representation, the representation must be zero dimensional.

In fact, there are stronger statements than the absence of finite-dimensional representations. To any finitely generated A_n-module M, there is a corresponding subvariety Char( M) of called the 'characteristic variety' whose size roughly corresponds to the size of M (a finite-dimensional module would have zero-dimensional characteristic variety). Then Bernstein's inequality states that for M non-zero,

$\backslash dim(\backslash operatorname\{char\}(M))\backslash geq\; n$

An even stronger statement is Gabber's theorem, which states that Char( M) is a co-isotropic subvariety of for the natural symplectic form.

---

Positive characteristic The situation is considerably different in the case of a Weyl algebra over a field of characteristic .

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

---

Generalizations The ideals and automorphisms of $A\_1$ have been well-studied. The moduli space for its right ideal is known. However, the case for $A\_n$ is considerably harder and is related to the Jacobian conjecture.

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

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

---

Affine varieties Weyl algebras also generalize in the case of algebraic varieties. Consider a polynomial ring

$R\; =\; \backslash frac\{\backslash mathbb\{C\}x\_1,\backslash ldots,x\_n\}\{I\}.$

Then a differential operator is defined as a composition of $\backslash mathbb\{C\}$-linear derivations of $R$. This can be described explicitly as the quotient ring

$\backslash text\{Diff\}(R)\; =\; \backslash frac\{\backslash \{\; D\; \backslash in\; A\_n\backslash colon\; D(I)\; \backslash subseteq\; I\; \backslash \}\}\{\; I\backslash cdot\; A\_n\}.$

---

See also

• Jacobian conjecture
• Dixmier conjecture

---

Notes

• S. C. Coutinho (1995). 9780521551199, Cambridge University Press. ISBN 9780521551199
• J. Helmstetter (2025). 9783764386054, Birkhäuser. ISBN 9783764386054
• N.P. Landsman (2025). 9780444515605, Elsevier. ISBN 9780444515605
• P. Lounesto (2025). 9780817635251, Springer Science & Business Media. ISBN 9780817635251
• A. Micali (1992). 9780792316237, Springer Science & Business Media. ISBN 9780792316237
• Will Traves (2025). 9780817648756, Birkhäuse. ISBN 9780817648756
• Tsit Yuen Lam (2025). 9780387953250, Springer. ISBN 9780387953250

Post Comment