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 ${\displaystyle F}$ be a field, and let ${\displaystyle F[x]}$ be the ring of polynomials in one variable with coefficients in ${\displaystyle F}$. Then the corresponding Weyl algebra consists of differential operators of form

${\displaystyle f_{m}(x)\partial _{x}^{m}+f_{m-1}(x)\partial _{x}^{m-1}+\cdots +f_{1}(x)\partial _{x}+f_{0}(x)}$

This is the first Weyl algebra ${\displaystyle A_{1}}$. The n-th Weyl algebra ${\displaystyle A_{n}}$ are constructed similarly.

Alternatively, ${\displaystyle A_{1}}$ can be constructed as the quotient of the free algebra on two generators, q and p, by the ideal generated by ${\displaystyle ([p,q]-1)}$. Similarly, ${\displaystyle A_{n}}$ is obtained by quotienting the free algebra on 2n generators by the ideal generated by$${\displaystyle ([p_{i},q_{j}]-\delta _{i,j}),\quad \forall i,j=1,\dots ,n}$$where ${\displaystyle \delta _{i,j}}$ is the Kronecker delta.

More generally, let ${\displaystyle (R,\Delta )}$ be a partial differential ring with commuting derivatives ${\displaystyle \Delta =\lbrace \partial _{1},\ldots ,\partial _{m}\rbrace }$. The Weyl algebra associated to ${\displaystyle (R,\Delta )}$ is the noncommutative ring ${\displaystyle R[\partial _{1},\ldots ,\partial _{m}]}$ satisfying the relations ${\displaystyle \partial _{i}r=r\partial _{i}+\partial _{i}(r)}$ for all ${\displaystyle r\in R}$. The previous case is the special case where ${\displaystyle R=F[x_{1},\ldots ,x_{n}]}$ and ${\displaystyle \Delta =\lbrace \partial _{x_{1}},\ldots ,\partial _{x_{n}}\rbrace }$ where ${\displaystyle F}$ is a field.

This article discusses only the case of ${\displaystyle A_{n}}$ with underlying field ${\displaystyle F}$ 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.

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 ${\displaystyle (q_{1},p_{1},\dots ,q_{n},p_{n})}$. These coordinates satisfy the Poisson bracket relations:$${\displaystyle \{q_{i},q_{j}\}=0,\quad \{p_{i},p_{j}\}=0,\quad \{q_{i},p_{j}\}=\delta _{ij}.}$$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:$${\displaystyle [{\hat {q}}_{i},{\hat {q}}_{j}]=0,\quad [{\hat {p}}_{i},{\hat {p}}_{j}]=0,\quad [{\hat {q}}_{i},{\hat {p}}_{j}]=i\hbar \delta _{ij},}$$where ${\displaystyle [\cdot ,\cdot ]}$ denotes the commutator. Here, ${\displaystyle {\hat {q}}_{i}}$ and ${\displaystyle {\hat {p}}_{i}}$ are the operators corresponding to ${\displaystyle q_{i}}$ and ${\displaystyle p_{i}}$ respectively. Erwin Schrödinger proposed in 1926 the following:^[1]

• ${\displaystyle {\hat {q_{j}}}}$ with multiplication by ${\displaystyle x_{j}}$.
• ${\displaystyle {\hat {p}}_{j}}$ with ${\displaystyle -i\hbar \partial _{x_{j}}}$.

With this identification, the canonical commutation relation holds.

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

The Weyl algebra ${\displaystyle A_{n}}$ can be concretely constructed as a representation.

In the differential operator representation, similar to Schrödinger's canonical quantization, let ${\displaystyle q_{j}}$ be represented by multiplication on the left by ${\displaystyle x_{j}}$, and let ${\displaystyle p_{j}}$ be represented by differentiation on the left by ${\displaystyle \partial _{x_{j}}}$.

In the matrix representation, similar to the matrix mechanics, ${\displaystyle A_{1}}$ is represented by^[2]$${\displaystyle P={\begin{bmatrix}0&1&0&0&\cdots \\0&0&2&0&\cdots \\0&0&0&3&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}},\quad Q={\begin{bmatrix}0&0&0&0&\ldots \\1&0&0&0&\cdots \\0&1&0&0&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}}}$$

${\displaystyle A_{n}}$ 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 2n) equipped with a symplectic form ω. Define the Weyl algebra W(V) to be

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

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

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

${\displaystyle A_{n}}$ is also a quotient of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the central element of the Heisenberg algebra (namely [q, p]) equal to the unit of the universal enveloping algebra (called 1 above).

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)

${\displaystyle a_{1}\cdots a_{n}\mapsto {\frac {1}{n!}}\sum _{\sigma \in S_{n}}a_{\sigma (1)}\otimes \cdots \otimes a_{\sigma (n)}~.}$

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 quantization (if for the latter one restricts to polynomial functions), but the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication.

Stated in another way, let the Moyal star product be denoted ${\displaystyle f\star g}$, then the Weyl algebra is isomorphic to ${\displaystyle (\mathbb {C} [x_{1},\dots ,x_{n}],\star )}$.^[3]

In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra, which is also referred to as the orthogonal Clifford algebra.^[4][5]

The Weyl algebra is also referred to as the symplectic Clifford algebra.^[4][5][6] Weyl algebras represent for symplectic bilinear forms the same structure that Clifford algebras represent for non-degenerate symmetric bilinear forms.^[6]

The Weyl algebra can be constructed as a D-module.^[7] Specifically, the Weyl algebra corresponding to the polynomial ring ${\displaystyle R[x_{1},...,x_{n}]}$ with its usual partial differential structure is precisely equal to Grothendieck's ring of differential operations ${\displaystyle D_{\mathbb {A} _{R}^{n}/R}}$.^[7]

More generally, let ${\displaystyle X}$ be a smooth scheme over a ring ${\displaystyle R}$. Locally, ${\displaystyle X\to R}$ factors as an étale cover over some ${\displaystyle \mathbb {A} _{R}^{n}}$ equipped with the standard projection.^[8] Because "étale" means "(flat and) possessing null cotangent sheaf",^[9] this means that every D-module over such a scheme can be thought of locally as a module over the ${\displaystyle n^{\text{th}}}$ Weyl algebra.

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

• ${\displaystyle D_{R}^{0}=R}$
• ${\displaystyle D_{R}^{k}=\left\{d\in \operatorname {End} _{S}(R):[d,a]\in D_{R}^{k-1}{\text{ for all }}a\in R\right\}.}$

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

In the case ${\displaystyle R=S[x_{1},...,x_{n}]}$, the ring of differential operators of order ${\displaystyle \leq n}$ presents similarly as in the special case ${\displaystyle S=\mathbb {C} }$ but for the added consideration of "divided power operators"; these are operators corresponding to those in the complex case which stabilize ${\displaystyle \mathbb {Z} [x_{1},...,x_{n}]}$, but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit ${\displaystyle D_{\mathbb {A} _{\mathbb {Z} }^{n}/\mathbb {Z} }}$. One such example is the operator ${\displaystyle \partial _{x_{1}}^{[p]}:x_{1}^{N}\mapsto {N \choose p}x_{1}^{N-p}}$.

Explicitly, a presentation is given by

${\displaystyle D_{S[x_{1},\dots ,x_{\ell }]/S}^{n}=S\langle x_{1},\dots ,x_{\ell },\{\partial _{x_{i}},\partial _{x_{i}}^{[2]},\dots ,\partial _{x_{i}}^{[n]}\}_{1\leq i\leq \ell }\rangle }$

with the relations

${\displaystyle [x_{i},x_{j}]=[\partial _{x_{i}}^{[k]},\partial _{x_{j}}^{[m]}]=0}$

${\displaystyle [\partial _{x_{i}}^{[k]},x_{j}]=\left\{{\begin{matrix}\partial _{x_{i}}^{[k-1]}&{\text{if }}i=j\\0&{\text{if }}i\neq j\end{matrix}}\right.}$

${\displaystyle \partial _{x_{i}}^{[k]}\partial _{x_{i}}^{[m]}={k+m \choose k}\partial _{x_{i}}^{[k+m]}~~~~~{\text{when }}k+m\leq n}$

where ${\displaystyle \partial _{x_{i}}^{[0]}=1}$ by convention. The Weyl algebra then consists of the limit of these algebras as ${\displaystyle n\to \infty }$.^{[10]: Ch. IV.16.II}

When ${\displaystyle S}$ is a field of characteristic 0, then ${\displaystyle D_{R}^{1}}$ is generated, as an ${\displaystyle R}$-module, by 1 and the ${\displaystyle S}$-derivations of ${\displaystyle R}$. Moreover, ${\displaystyle D_{R}}$ is generated as a ring by the ${\displaystyle R}$-subalgebra ${\displaystyle D_{R}^{1}}$. In particular, if ${\displaystyle S=\mathbb {C} }$ and ${\displaystyle R=\mathbb {C} [x_{1},...,x_{n}]}$, then ${\displaystyle D_{R}^{1}=R+\sum _{i}R\partial _{x_{i}}}$. As mentioned, ${\displaystyle A_{n}=D_{R}}$.^[11]

Many properties of ${\displaystyle A_{1}}$ apply to ${\displaystyle A_{n}}$ with essentially similar proofs, since the different dimensions commute.

In particular, ${\textstyle [q,q^{m}p^{n}]=-nq^{m}p^{n-1}}$ and ${\textstyle [p,q^{m}p^{n}]=mq^{m-1}p^{n}}$.

This allows ${\displaystyle A_{1}}$ to be a graded algebra, where the degree of ${\displaystyle \sum _{m,n}c_{m,n}q^{m}p^{n}}$ is ${\displaystyle \max(m+n)}$ among its nonzero monomials. The degree is similarly defined for ${\displaystyle A_{n}}$.

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

That is, any derivation ${\textstyle D}$ is equal to ${\textstyle [\cdot ,f]}$ for some ${\textstyle f\in A_{n}}$; any ${\textstyle f\in A_{n}}$ yields a derivation ${\textstyle [\cdot ,f]}$; if ${\textstyle f,f'\in A_{n}}$ satisfies ${\textstyle [\cdot ,f]=[\cdot ,f']}$, then ${\textstyle f-f'\in F}$.

The proof is similar to computing the potential function for a conservative polynomial vector field on the plane.^[16]

In the case that the ground field F has characteristic zero, the nth Weyl algebra is a simple Noetherian domain.^[17] It has global dimension n, in contrast to the ring it deforms, Sym(V), which has global dimension 2n.

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 [q,p] = 1).

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

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 V × V^∗ called the 'characteristic variety'^{[clarification needed]} whose size roughly corresponds to the size^{[clarification needed]} of M (a finite-dimensional module would have zero-dimensional characteristic variety). Then Bernstein's inequality states that for M non-zero,

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

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

The situation is considerably different in the case of a Weyl algebra over a field of characteristic p > 0.

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.

The ideals and automorphisms of ${\displaystyle A_{1}}$ have been well-studied.^[18][19] The moduli space for its right ideal is known.^[20] However, the case for ${\displaystyle A_{n}}$ is considerably harder and is related to the Jacobian conjecture.^[21]

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.

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

${\displaystyle R={\frac {\mathbb {C} [x_{1},\ldots ,x_{n}]}{I}}.}$

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

${\displaystyle {\text{Diff}}(R)={\frac {\{D\in A_{n}\colon D(I)\subseteq I\}}{I\cdot A_{n}}}.}$

• Jacobian conjecture
• Dixmier conjecture

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